// This file is in Magma syntax.
// It deals with matrix calculations.
// The file cs-surface-presentations_magma.txt deals with calculations
// involving the presentations of the various groups.

// The fields involved:
L<z>:=CyclotomicField(12); // z is a primitive 12-th root of 1.
K<r>:=sub< L | z+1/z >;    // K is \Q(\sqrt(3))
                           // r=z+1/z is a square root of 3, positive
                           // if $z=e^{2\pi i/12}$. Note that r = 2*z-z^3;
RR:=RealField(20);
hKR:=hom<K->RR | Sqrt(3)>; // Embedding of $K$ in $\R$.
hLL:=hom<L->L | z^-1 >;    // Complex conjugation on L.

MatsL:=MatrixRing(L,3);
ID3:=MatsL!1;

// Calculations for Section 1.1:
// *****************************
// Here is a form we can use. The "emptyset" in the name is a reference
// to the naming scheme Cartwright and Steger used in their work on fake
// projective planes. The determinant of the form is 1. Its eigenvalues
// are 1 and -r+\sqrt2 and -r-sqrt2, and so are all positive if
// r is the negative square root of 3. If we take r to be the positive
// square root of 3, then one eigenvalue of Femptyset is positive, and
// the other two are negative.

// In the write-up, it is multiplied by -1 so that $B^2_\C(F)=\{[z]:F(z)<0\}$.

Femptyset:=MatsL![
-r-1,  1,0,
   1,1-r,0,
   0,  0,1];

// Inverse of Femptyset
FemptysetI:=MatsL![
1-r,  -1,0,
 -1,-1-r,0,
  0,   0,1];

// We will mostly use the following diagonal form. Notice that 
// its (3,3) entry is negative when r is the positive square root of 3:

Fdiag:=MatsL![1,0,0,
              0,1,0,
              0,0,1-r];

// Here is the inverse of Fdiag:

FdiagI:=MatsL![1,0,0,
           0,1,0,
           0,0,-(1+r)/2];

// The function starmat calculates the complex conjugate 
// of the transpose of a matrix in MatsL.

starmat:=function(matvar);
return MatsL![hLL(matvar[1][1]),hLL(matvar[2][1]),hLL(matvar[3][1]),
              hLL(matvar[1][2]),hLL(matvar[2][2]),hLL(matvar[3][2]),
              hLL(matvar[1][3]),hLL(matvar[2][3]),hLL(matvar[3][3])];
end function;

// We translate between the two forms using the following matrix:

GAMMA0:=MatsL![
1,  0,0,
1,1-r,0,
0,  0,1];

// Here is inverse of GAMMA0:
GAMMA0I:=MatsL![
      1,       0,0,
(r+1)/2,(-r-1)/2,0,
      0,       0,1];

// Checking that we can get from one form to the other using GAMMA0:

print "Femptyset - (1/(1-r))*starmat(GAMMA0))*Fdiag*GAMMA0 = ",
Femptyset - (1/(1-r))*starmat(GAMMA0)*Fdiag*GAMMA0;

// absFdiag equals Fdiag multiplied by the diagonal matrix Fstandard having diagonal entries 1,1,-1.
// If a matrix g is unitary with respect to Fdiag, then h = absFdiag^{1/2}g absFdiag^{-1/2} is 
// unitary with respect to Fstandard, and the squared Hilbert-Schmidt norm of h
// (i.e., \sum_{i,j}|h_{i,j}|^2) equals the trace of h^*h. This is calculated
// by the function hsnormfn defined next. 

absFdiag:=MatsL![1,0,0,
                 0,1,0,
                 0,0,r-1];

absFdiagI:=MatsL![1,0,0,
                  0,1,0,
                  0,0,(r+1)/2];

hsnormfn:=function(matvar);
return hKR(Trace(starmat(matvar)*absFdiag*matvar*absFdiagI));
end function;

// Here are matrices which are unitary with respect to
// Femptyset, and which have entries in $\Z[z]$. When we conjugate
// them by GAMMA0, we get matrices unitary with respect to Fdiag
// but which have entries in $(r-1)^{-1}\Z[z]$. Conversely, conjugating 
// a matrix which is unitary with respect to Fdiag by GAMMA0I transforms
// it into a matrix which are unitary with respect to the form Femptyset.

// Here are the matrices which are called $u'$, $v'$ and $b'$ in the write-up.
UD:=MatsL![
z^3+z^2-z,1-z,0,
z^3+z^2-1,z-z^3,0,
0,0,1];

VD:=MatsL![
        z^3,0,0,
z^3+z^2-z-1,1,0,
          0,0,1];

BD:=MatsL![
               1,          0,       0,
-2*z^3-z^2+2*z+2,z^3+z^2-z-1,-z^3-z^2,
           z^2+z,     -z^3-1,-z^3+z+1];

// We also set JD:=(UD*VD)^2;

JD:=MatsL![
z,0,0,
0,z,0,
0,0,1];

UDI:=UD^-1;
VDI:=VD^-1;
JDI:=JD^-1;
BDI:=BD^-1;

// The group $\bar\Gamma$ is generated by the matrices U, V and B (modulo powers of z)
// The matrix J is just introduced to make the expressions for the elements of 
// $K=\langle u,v\rangle$ neater.

// Here are the matrices called $u$, $v$, $j$ and~$b$ in the write-up.";
U:=GAMMA0*UD*GAMMA0I;
V:=GAMMA0*VD*GAMMA0I;
J:=GAMMA0*JD*GAMMA0I;
B:=GAMMA0*BD*GAMMA0I;

UI:=U^-1;
VI:=V^-1;
JI:=J^-1;
BI:=B^-1;

// The relations from our presentation of $\bar\Gamma$.
// The following are zero.
U^3 - ID3;
V^4 - ID3;
(U*V)^2 - (V*U)^2;
B^3 - ID3;
B*V - V*B;
(B*U*V)^3 - (1/z)*ID3;
(B*U*V*U)^2*V - (1/z)*ID3;
print "Finished checking that the matrices U, V and B";
print "satisfy the claimed relations.";

// We next form a list of the 288 elements of $K$. To give
// simple words for them, we use the subgroup $J=\langle j\rangle$
// of $K$, which has order 12 and is the center of $K$. The following
// list of 24 elements $t_1,\ldots,t_{24}$ of~$K$ is a transversal 
// for $J$ in $K$. That is, each element of $K$ can be written as a 
// product $t_a*J^b$, where $a\in\{1,\ldots,24\}$ and $b\in\{0,\ldots,11\}$.
// The list Jtransversal is a modification of the list found using the 
// Magma command Transversal(Gp,Subgp); (see the file cs-surface-presentations_magma.txt).
Jtransversal:=[MatsL |
ID3,
V,
VI,
U,
UI,
V^2,
V*U,
V*UI,
VI*U,
U*V,
UI*V,
U*VI,
V^2*U,
V^2*UI,
V*U*VI,
V*UI*V,
VI*U*VI,
U*V^2,
U*VI*U,
V^2*U*VI,
VI*U*V^2,
U*V^2*UI,
UI*V^2*U,
V*UI*V^2*U];

kklist:=[MatsL | Jtransversal[i1]*J^i2 : i1 in [1 .. 24], i2 in [0 .. 11]];

kklistI:=[MatsL | ];
for ii in [1 .. #kklist] do
 kklistI:=Append(kklistI,kklist[ii]^-1);
end for;

// The following function takes a matrix k in $K$, and finds the index i1
// such that k=kklist[i1+1]. If the matrix is not in $K$, it returns -1.

kkindex:=function(matvar);
 index:=-1;
 i1:=0;
 indexfound:=0;
 while ((i1 le 287) and (indexfound eq 0)) do
  if (matvar eq kklist[i1+1]) then 
   index:=i1; 
   indexfound:=1;
  end if;
  i1+:=1;
 end while;
return index;
end function;

Jtransversalwordlist:=[
[],
["V"],
["VI"],
["U"],
["UI"],
["V","V"],
["V","U"],
["V","UI"],
["VI","U"],
["U","V"],
["UI","V"],
["U","VI"],
["V","V","U"],
["V","V","UI"],
["V","U","VI"],
["V","UI","V"],
["VI","U","VI"],
["U","V","V"],
["U","VI","U"],
["V","V","U","VI"],
["VI","U","V","V"],
["U","V","V","UI"],
["UI","V","V","U"],
["V","UI","V","V","U"]];

Jwordlist:=[
[],
["J"],
["J","J"],
["J","J","J"],
["J","J","J","J"],
["J","J","J","J","J"],
["J","J","J","J","J","J"],
["J","J","J","J","J","J","J"],
["J","J","J","J","J","J","J","J"],
["J","J","J","J","J","J","J","J","J"],
["J","J","J","J","J","J","J","J","J","J"],
["J","J","J","J","J","J","J","J","J","J","J"]];

kkwordlist:=[Jtransversalwordlist[i1] cat Jwordlist[i2] : i1 in [1 .. 24], i2 in [1 .. 12]];
print "The number of elements in kklist is",#kklist;

for ii in [1 .. #kklist] do
 for jj in [ii+1 .. #kklist] do
  tmpmat:=kklist[jj]*kklist[ii]^(-1);
  if (tmpmat eq tmpmat[1][1]*ID3) then print "kklist[",ii,"] equals kklist[",jj,"]"; end if;
 end for;
end for;
print "finished checking that the 288 elements of kklist are distinct";

// End of calculations for section 1.1.
// ************************************

// Calculations for section 1.3.
// *****************************

// Calculations for Proposition 1:
// *******************************
// // Here is a routine for finding the elements of finite 
// // order in $K\cup bK\cup bu^{-1}bK$:
// 
// firsteltlist:=[MatsL | ID3,B,B*UI*B];
// firsteltwordlist:=[[],["B"],["B*UI*B"]];
// 
// for d in [2 .. 24] do
//  if IsDivisibleBy(24,d) then
//   print "\n";
//   print "order",d,"list:=[MatsL |";
//   for idx in [1,2,3] do
//    for ii in [1 .. 288] do
//     testmat:=(firsteltlist[idx]*kklist[ii])^d;
//     if (testmat eq testmat[1][1]*ID3) then
//      d_is_order:=1;
//      if IsDivisibleBy(d,2) then 
//       d2:=IntegerRing()!(d/2);
//       testmat2:=(firsteltlist[idx]*kklist[ii])^d2;
//       if (testmat2 eq testmat2[1][1]*ID3) then d_is_order:=0; end if;
//      end if;
//      if IsDivisibleBy(d,3) then 
//       d3:=IntegerRing()!(d/3);
//       testmat3:=(firsteltlist[idx]*kklist[ii])^d3;
//       if (testmat3 eq testmat3[1][1]*ID3) then d_is_order:=0; end if;
//      end if;
//      if (d_is_order eq 1) then print firsteltwordlist[idx],"*",kkwordlist[ii],","; end if;
//     end if;
//    end for;
//   end for;
//   print "];";
//  end if;
// end for;
// 
// order2list:=[MatsL|
// V*V,
// U*V*V*UI,
// UI*V*V*U,
// J*J*J*J*J*J,
// V*V*J*J*J*J*J*J,
// U*V*V*UI*J*J*J*J*J*J,
// UI*V*V*U*J*J*J*J*J*J,
// B*UI*B*UI];
// 
// order3list:=[MatsL|
// U,
// UI,
// V*U*VI,
// V*V*UI*J,
// V*UI*V*J,
// V*V*U*J*J*J,
// VI*U*VI*J*J*J,
// U*V*V*J*J*J,
// J*J*J*J,
// U*J*J*J*J,
// UI*J*J*J*J,
// V*U*VI*J*J*J*J,
// V*V*UI*J*J*J*J*J,
// V*UI*V*J*J*J*J*J,
// V*V*U*J*J*J*J*J*J*J,
// VI*U*VI*J*J*J*J*J*J*J,
// U*V*V*J*J*J*J*J*J*J,
// J*J*J*J*J*J*J*J,
// U*J*J*J*J*J*J*J*J,
// UI*J*J*J*J*J*J*J*J,
// V*U*VI*J*J*J*J*J*J*J*J,
// V*V*UI*J*J*J*J*J*J*J*J*J,
// V*UI*V*J*J*J*J*J*J*J*J*J,
// V*V*U*J*J*J*J*J*J*J*J*J*J*J,
// VI*U*VI*J*J*J*J*J*J*J*J*J*J*J,
// U*V*V*J*J*J*J*J*J*J*J*J*J*J,
// B,
// B*V*U,
// B*U*V,
// B*V*V*U*VI,
// B*VI*U*V*V,
// B*U*J,
// B*V*U*VI*J,
// B*VI*J*J,
// B*V*V*UI*J*J,
// B*V*UI*V*J*J,
// B*UI*B*VI,
// B*UI*B*V*U,
// B*UI*B*U*V,
// B*UI*B*U*VI*U*J,
// B*UI*B*V*UI*J*J*J*J*J*J*J*J*J*J*J,
// B*UI*B*UI*V*J*J*J*J*J*J*J*J*J*J*J];
// 
// order4list:=[MatsL|
// V,
// VI,
// VI*U*J,
// U*VI*J,
// V*UI*J*J,
// UI*V*J*J,
// J*J*J,
// V*J*J*J,
// VI*J*J*J,
// V*V*J*J*J,
// U*V*V*UI*J*J*J,
// UI*V*V*U*J*J*J,
// VI*U*J*J*J*J,
// U*VI*J*J*J*J,
// V*UI*J*J*J*J*J,
// UI*V*J*J*J*J*J,
// V*J*J*J*J*J*J,
// VI*J*J*J*J*J*J,
// VI*U*J*J*J*J*J*J*J,
// U*VI*J*J*J*J*J*J*J,
// V*UI*J*J*J*J*J*J*J*J,
// UI*V*J*J*J*J*J*J*J*J,
// J*J*J*J*J*J*J*J*J,
// V*J*J*J*J*J*J*J*J*J,
// VI*J*J*J*J*J*J*J*J*J,
// V*V*J*J*J*J*J*J*J*J*J,
// U*V*V*UI*J*J*J*J*J*J*J*J*J,
// UI*V*V*U*J*J*J*J*J*J*J*J*J,
// VI*U*J*J*J*J*J*J*J*J*J*J,
// U*VI*J*J*J*J*J*J*J*J*J*J,
// V*UI*J*J*J*J*J*J*J*J*J*J*J,
// UI*V*J*J*J*J*J*J*J*J*J*J*J,
// B*UI,
// B*VI*U*J*J,
// B*U*VI*J*J,
// B*V*UI*J*J*J,
// B*UI*V*J*J*J,
// B*V*V*U*J*J*J*J*J*J*J*J*J*J*J,
// B*VI*U*VI*J*J*J*J*J*J*J*J*J*J*J,
// B*U*V*V*J*J*J*J*J*J*J*J*J*J*J,
// B*UI*B*J,
// B*UI*B*U*J*J*J*J*J*J*J*J*J*J*J];
// 
// order6list:=[MatsL|
// V*V*U*J,
// VI*U*VI*J,
// U*V*V*J,
// J*J,
// U*J*J,
// UI*J*J,
// V*V*J*J,
// V*U*VI*J*J,
// U*V*V*UI*J*J,
// UI*V*V*U*J*J,
// V*V*UI*J*J*J,
// V*UI*V*J*J*J,
// V*V*J*J*J*J,
// U*V*V*UI*J*J*J*J,
// UI*V*V*U*J*J*J*J,
// V*V*U*J*J*J*J*J,
// VI*U*VI*J*J*J*J*J,
// U*V*V*J*J*J*J*J,
// U*J*J*J*J*J*J,
// UI*J*J*J*J*J*J,
// V*U*VI*J*J*J*J*J*J,
// V*V*UI*J*J*J*J*J*J*J,
// V*UI*V*J*J*J*J*J*J*J,
// V*V*J*J*J*J*J*J*J*J,
// U*V*V*UI*J*J*J*J*J*J*J*J,
// UI*V*V*U*J*J*J*J*J*J*J*J,
// V*V*U*J*J*J*J*J*J*J*J*J,
// VI*U*VI*J*J*J*J*J*J*J*J*J,
// U*V*V*J*J*J*J*J*J*J*J*J,
// J*J*J*J*J*J*J*J*J*J,
// U*J*J*J*J*J*J*J*J*J*J,
// UI*J*J*J*J*J*J*J*J*J*J,
// V*V*J*J*J*J*J*J*J*J*J*J,
// V*U*VI*J*J*J*J*J*J*J*J*J*J,
// U*V*V*UI*J*J*J*J*J*J*J*J*J*J,
// UI*V*V*U*J*J*J*J*J*J*J*J*J*J,
// V*V*UI*J*J*J*J*J*J*J*J*J*J*J,
// V*UI*V*J*J*J*J*J*J*J*J*J*J*J,
// B*U,
// B*V*V,
// B*V*U*VI,
// B*V*U*J,
// B*U*V*J,
// B*V*V*UI*J,
// B*V*UI*V*J,
// B*V*V*U*VI*J,
// B*VI*U*V*V*J,
// B*V*J*J,
// B*UI*B,
// B*UI*B*U,
// B*UI*B*V*J,
// B*UI*B*VI*U*J*J,
// B*UI*B*U*VI*J*J,
// B*UI*B*V*J*J*J*J*J*J*J*J*J*J,
// B*UI*B*VI*U*J*J*J*J*J*J*J*J*J*J*J,
// B*UI*B*U*VI*J*J*J*J*J*J*J*J*J*J*J];
// 
// order8list:=[MatsL|
// V*UI*V*V*U,
// V*U*J,
// U*V*J,
// V*V*U*VI*J,
// VI*U*V*V*J,
// U*VI*U*J*J,
// V*UI*V*V*U*J*J*J,
// V*U*J*J*J*J,
// U*V*J*J*J*J,
// V*V*U*VI*J*J*J*J,
// VI*U*V*V*J*J*J*J,
// U*VI*U*J*J*J*J*J,
// V*UI*V*V*U*J*J*J*J*J*J,
// V*U*J*J*J*J*J*J*J,
// U*V*J*J*J*J*J*J*J,
// V*V*U*VI*J*J*J*J*J*J*J,
// VI*U*V*V*J*J*J*J*J*J*J,
// U*VI*U*J*J*J*J*J*J*J*J,
// V*UI*V*V*U*J*J*J*J*J*J*J*J*J,
// V*U*J*J*J*J*J*J*J*J*J*J,
// U*V*J*J*J*J*J*J*J*J*J*J,
// V*V*U*VI*J*J*J*J*J*J*J*J*J*J,
// VI*U*V*V*J*J*J*J*J*J*J*J*J*J,
// U*VI*U*J*J*J*J*J*J*J*J*J*J*J,
// B*J,
// B*V*J,
// B*VI*J,
// B*V*V*J,
// B*U*J*J,
// B*V*U*VI*J*J,
// B*V*V*UI*J*J*J,
// B*V*UI*V*J*J*J,
// B*V*U*J*J*J*J*J*J*J*J*J*J*J,
// B*U*V*J*J*J*J*J*J*J*J*J*J*J,
// B*V*V*U*VI*J*J*J*J*J*J*J*J*J*J*J,
// B*VI*U*V*V*J*J*J*J*J*J*J*J*J*J*J,
// B*UI*B*V*V*J,
// B*UI*B*U*V*V*UI*J,
// B*UI*B*UI*V*V*U*J,
// B*UI*B*V*UI*V*J*J*J*J*J*J*J*J*J,
// B*UI*B*V*V*U*J*J*J*J*J*J*J*J*J*J*J,
// B*UI*B*U*V*V*J*J*J*J*J*J*J*J*J*J*J];
// 
// order12list:=[MatsL|
// V*UI,
// VI*U,
// UI*V,
// U*VI,
// V*V*U,
// V*V*UI,
// V*UI*V,
// VI*U*VI,
// U*V*V,
// J,
// V*J,
// VI*J,
// U*J,
// UI*J,
// V*V*J,
// V*UI*J,
// UI*V*J,
// V*U*VI*J,
// U*V*V*UI*J,
// UI*V*V*U*J,
// V*J*J,
// VI*J*J,
// VI*U*J*J,
// U*VI*J*J,
// V*V*U*J*J,
// V*V*UI*J*J,
// V*UI*V*J*J,
// VI*U*VI*J*J,
// U*V*V*J*J,
// U*J*J*J,
// UI*J*J*J,
// V*UI*J*J*J,
// VI*U*J*J*J,
// UI*V*J*J*J,
// U*VI*J*J*J,
// V*U*VI*J*J*J,
// V*J*J*J*J,
// VI*J*J*J*J,
// V*UI*J*J*J*J,
// UI*V*J*J*J*J,
// V*V*U*J*J*J*J,
// V*V*UI*J*J*J*J,
// V*UI*V*J*J*J*J,
// VI*U*VI*J*J*J*J,
// U*V*V*J*J*J*J,
// J*J*J*J*J,
// V*J*J*J*J*J,
// VI*J*J*J*J*J,
// U*J*J*J*J*J,
// UI*J*J*J*J*J,
// V*V*J*J*J*J*J,
// VI*U*J*J*J*J*J,
// U*VI*J*J*J*J*J,
// V*U*VI*J*J*J*J*J,
// U*V*V*UI*J*J*J*J*J,
// UI*V*V*U*J*J*J*J*J,
// V*UI*J*J*J*J*J*J,
// VI*U*J*J*J*J*J*J,
// UI*V*J*J*J*J*J*J,
// U*VI*J*J*J*J*J*J,
// V*V*U*J*J*J*J*J*J,
// V*V*UI*J*J*J*J*J*J,
// V*UI*V*J*J*J*J*J*J,
// VI*U*VI*J*J*J*J*J*J,
// U*V*V*J*J*J*J*J*J,
// J*J*J*J*J*J*J,
// V*J*J*J*J*J*J*J,
// VI*J*J*J*J*J*J*J,
// U*J*J*J*J*J*J*J,
// UI*J*J*J*J*J*J*J,
// V*V*J*J*J*J*J*J*J,
// V*UI*J*J*J*J*J*J*J,
// UI*V*J*J*J*J*J*J*J,
// V*U*VI*J*J*J*J*J*J*J,
// U*V*V*UI*J*J*J*J*J*J*J,
// UI*V*V*U*J*J*J*J*J*J*J,
// V*J*J*J*J*J*J*J*J,
// VI*J*J*J*J*J*J*J*J,
// VI*U*J*J*J*J*J*J*J*J,
// U*VI*J*J*J*J*J*J*J*J,
// V*V*U*J*J*J*J*J*J*J*J,
// V*V*UI*J*J*J*J*J*J*J*J,
// V*UI*V*J*J*J*J*J*J*J*J,
// VI*U*VI*J*J*J*J*J*J*J*J,
// U*V*V*J*J*J*J*J*J*J*J,
// U*J*J*J*J*J*J*J*J*J,
// UI*J*J*J*J*J*J*J*J*J,
// V*UI*J*J*J*J*J*J*J*J*J,
// VI*U*J*J*J*J*J*J*J*J*J,
// UI*V*J*J*J*J*J*J*J*J*J,
// U*VI*J*J*J*J*J*J*J*J*J,
// V*U*VI*J*J*J*J*J*J*J*J*J,
// V*J*J*J*J*J*J*J*J*J*J,
// VI*J*J*J*J*J*J*J*J*J*J,
// V*UI*J*J*J*J*J*J*J*J*J*J,
// UI*V*J*J*J*J*J*J*J*J*J*J,
// V*V*U*J*J*J*J*J*J*J*J*J*J,
// V*V*UI*J*J*J*J*J*J*J*J*J*J,
// V*UI*V*J*J*J*J*J*J*J*J*J*J,
// VI*U*VI*J*J*J*J*J*J*J*J*J*J,
// U*V*V*J*J*J*J*J*J*J*J*J*J,
// J*J*J*J*J*J*J*J*J*J*J,
// V*J*J*J*J*J*J*J*J*J*J*J,
// VI*J*J*J*J*J*J*J*J*J*J*J,
// U*J*J*J*J*J*J*J*J*J*J*J,
// UI*J*J*J*J*J*J*J*J*J*J*J,
// V*V*J*J*J*J*J*J*J*J*J*J*J,
// VI*U*J*J*J*J*J*J*J*J*J*J*J,
// U*VI*J*J*J*J*J*J*J*J*J*J*J,
// V*U*VI*J*J*J*J*J*J*J*J*J*J*J,
// U*V*V*UI*J*J*J*J*J*J*J*J*J*J*J,
// UI*V*V*U*J*J*J*J*J*J*J*J*J*J*J,
// B*V,
// B*VI,
// B*U*V*V*UI,
// B*UI*V*V*U,
// B*U*VI*U*J,
// B*J*J,
// B*V*V*J*J,
// B*J*J*J,
// B*V*J*J*J,
// B*VI*J*J*J,
// B*V*V*J*J*J,
// B*V*V*U*J*J*J,
// B*VI*U*VI*J*J*J,
// B*U*V*V*J*J*J,
// B*UI*J*J*J*J,
// B*V*UI*V*V*U*J*J*J*J*J,
// B*U*V*V*UI*J*J*J*J*J*J,
// B*UI*V*V*U*J*J*J*J*J*J,
// B*U*VI*U*J*J*J*J*J*J*J,
// B*VI*U*J*J*J*J*J*J*J*J*J*J,
// B*U*VI*J*J*J*J*J*J*J*J*J*J,
// B*J*J*J*J*J*J*J*J*J*J*J,
// B*V*J*J*J*J*J*J*J*J*J*J*J,
// B*VI*J*J*J*J*J*J*J*J*J*J*J,
// B*V*V*J*J*J*J*J*J*J*J*J*J*J,
// B*V*UI*J*J*J*J*J*J*J*J*J*J*J,
// B*UI*V*J*J*J*J*J*J*J*J*J*J*J,
// B*V*UI*V*V*U*J*J*J*J*J*J*J*J*J*J*J,
// B*UI*B*V*UI,
// B*UI*B*UI*V,
// B*UI*B*VI*U*VI,
// B*UI*B*U*VI*U,
// B*UI*B*VI*J,
// B*UI*B*U*J,
// B*UI*B*UI*J,
// B*UI*B*V*U*VI*J,
// B*UI*B*V*V*UI*J*J,
// B*UI*B*VI*U*VI*J*J*J*J,
// B*UI*B*V*U*VI*J*J*J*J*J*J*J*J*J,
// B*UI*B*V*V*UI*J*J*J*J*J*J*J*J*J*J,
// B*UI*B*J*J*J*J*J*J*J*J*J*J*J,
// B*UI*B*UI*J*J*J*J*J*J*J*J*J*J*J,
// B*UI*B*V*U*J*J*J*J*J*J*J*J*J*J*J,
// B*UI*B*U*V*J*J*J*J*J*J*J*J*J*J*J];
// 
// order24list:=[MatsL|
// V*U,
// U*V,
// U*VI*U,
// V*V*U*VI,
// VI*U*V*V,
// U*VI*U*J,
// V*UI*V*V*U*J,
// V*U*J*J,
// U*V*J*J,
// V*V*U*VI*J*J,
// VI*U*V*V*J*J,
// V*UI*V*V*U*J*J,
// V*U*J*J*J,
// U*V*J*J*J,
// U*VI*U*J*J*J,
// V*V*U*VI*J*J*J,
// VI*U*V*V*J*J*J,
// U*VI*U*J*J*J*J,
// V*UI*V*V*U*J*J*J*J,
// V*U*J*J*J*J*J,
// U*V*J*J*J*J*J,
// V*V*U*VI*J*J*J*J*J,
// VI*U*V*V*J*J*J*J*J,
// V*UI*V*V*U*J*J*J*J*J,
// V*U*J*J*J*J*J*J,
// U*V*J*J*J*J*J*J,
// U*VI*U*J*J*J*J*J*J,
// V*V*U*VI*J*J*J*J*J*J,
// VI*U*V*V*J*J*J*J*J*J,
// U*VI*U*J*J*J*J*J*J*J,
// V*UI*V*V*U*J*J*J*J*J*J*J,
// V*U*J*J*J*J*J*J*J*J,
// U*V*J*J*J*J*J*J*J*J,
// V*V*U*VI*J*J*J*J*J*J*J*J,
// VI*U*V*V*J*J*J*J*J*J*J*J,
// V*UI*V*V*U*J*J*J*J*J*J*J*J,
// V*U*J*J*J*J*J*J*J*J*J,
// U*V*J*J*J*J*J*J*J*J*J,
// U*VI*U*J*J*J*J*J*J*J*J*J,
// V*V*U*VI*J*J*J*J*J*J*J*J*J,
// VI*U*V*V*J*J*J*J*J*J*J*J*J,
// U*VI*U*J*J*J*J*J*J*J*J*J*J,
// V*UI*V*V*U*J*J*J*J*J*J*J*J*J*J,
// V*U*J*J*J*J*J*J*J*J*J*J*J,
// U*V*J*J*J*J*J*J*J*J*J*J*J,
// V*V*U*VI*J*J*J*J*J*J*J*J*J*J*J,
// VI*U*V*V*J*J*J*J*J*J*J*J*J*J*J,
// V*UI*V*V*U*J*J*J*J*J*J*J*J*J*J*J,
// B*V*V*U,
// B*VI*U*VI,
// B*U*V*V,
// B*UI*J,
// B*VI*U*J,
// B*U*VI*J,
// B*V*UI*J*J,
// B*UI*V*J*J,
// B*UI*B*V*V,
// B*UI*B*V*V*U,
// B*UI*B*U*V*V,
// B*UI*B*U*V*V*UI,
// B*UI*B*UI*V*V*U,
// B*UI*B*V*UI*V*J*J*J*J*J*J*J*J*J*J];
// 
// 
// order2replist:=[MatsL |
// V*V,
// J*J*J*J*J*J,
// B*UI*B*UI];
// 
// order2repwordlist:=[
// ["V*V"],
// ["J*J*J*J*J*J"],
// ["B*UI*B*UI"]];
// 
// order3replist:=[MatsL |
// U,
// J*J*J*J,
// U*J*J*J*J,
// B*U*V];
// 
// order3repwordlist:=[
// ["U"],
// ["J*J*J*J"],
// ["U*J*J*J*J"],
// ["B*U*V"]];
// 
// order4replist:=[MatsL|
// V,
// J*J*J,
// V*J*J*J,
// V*V*J*J*J,
// B*UI];
// 
// order4repwordlist:=[
// ["V"],
// ["J*J*J"],
// ["V*J*J*J"],
// ["V*V*J*J*J"],
// ["B*UI"]];
// 
// order6replist:=[MatsL|
// J*J,
// V*V*J*J,
// V*V*U*J,
// V*V*U*J*J*J*J*J,
// B*V*V*UI*J,
// B*V*V];
// 
// order6repwordlist:=[
// ["J*J"],
// ["V*V*J*J"],
// ["V*V*U*J"],
// ["V*V*U*J*J*J*J*J"],
// ["B*V*V*UI*J"],
// ["B*V*V"]];
// 
// order8replist:=[MatsL|
// U*V*J,
// B*J,
// (B*J)^3];
// 
// order8repwordlist:=[
// ["U*V*J"],
// ["B*J"],
// ["(B*J)^3"]];
// 
// order12replist:=[MatsL|
// J,
// J*J*J*J*J,
// U*VI*J*J,
// U*VI*J*J*J,
// U*VI*J*J*J*J*J*J,
// U*VI*J*J*J*J*J*J*J*J*J*J*J,
// V*V*J,
// U*V*V,
// U*J,
// U*J*J*J,
// B*V,
// (B*V)^(-5)];
// 
// order12repwordlist:=[
// ["J"],
// ["J*J*J*J*J"],
// ["U*VI*J*J"],
// ["U*VI*J*J*J"],
// ["U*VI*J*J*J*J*J*J"],
// ["U*VI*J*J*J*J*J*J*J*J*J*J*J"],
// ["V*V*J"],
// ["U*V*V"],
// ["U*J"],
// ["U*J*J*J"],
// ["B*V"],
// ["(B*V)^(-5)"]];
// 
// order24replist:=[MatsL|
// U*V,
// V*U*J*J];
// 
// order24repwordlist:=[
// ["U*V"],
// ["V*U*J*J"]];
// 
// BUIB:=B*UI*B;
// 
// print "Confirming representatives for elements of order 12";
// print "The number of elements in order12list is",#order12list;
// for ii in [1 .. #order12list] do
//  conjugatefound:=0;
//  target:=order12list[ii]^-1;
//  jj:=1;
//  while ((jj le #order12replist) and (conjugatefound eq 0)) do
//   i1:=1;
//   while ((i1 le 288) and (conjugatefound eq 0)) do
//    conjmat:=kklist[i1];
//    testmat:=target*conjmat*order12replist[jj]*conjmat^-1;
//    if (testmat eq testmat[1][1]*ID3) then 
//     conjugatefound:=1;
//     print "List element",ii,"is conjugate to",order12repwordlist[jj],"using",kkwordlist[i1];
//    end if;
//    i1+:=1;
//   end while;
//   jj+:=1;
//  end while;
//  if (conjugatefound eq 0) then
//   target:=order12list[ii];
//   jj:=1;
//   while ((jj le #order12replist) and (conjugatefound eq 0)) do
//    i1:=1;
//    while ((i1 le 288) and (conjugatefound eq 0)) do
//     conjmat:=kklist[i1];
//     testmat:=target*conjmat*order12replist[jj]*conjmat^-1;
//     if (testmat eq testmat[1][1]*ID3) then 
//      conjugatefound:=1;
//      print "List element",ii,"is conjugate to the inverse of",order12repwordlist[jj],"using",kkwordlist[i1];
//     end if;
//     i1+:=1;
//    end while;
//    jj+:=1;
//   end while;
//  end if;
//  if (conjugatefound eq 0) then
//   target:=order12list[ii]^-1;
//   jj:=1;
//   while ((jj le #order12replist) and (conjugatefound eq 0)) do
//    i1:=1;
//    while ((i1 le 288) and (conjugatefound eq 0)) do
//     i2:=1;
//     while ((i2 le 288) and (conjugatefound eq 0)) do
//      conjmat:=kklist[i1]*B*kklist[i2];
//      testmat:=target*conjmat*order12replist[jj]*conjmat^-1;
//      if (testmat eq testmat[1][1]*ID3) then 
//       conjugatefound:=1;
//       print "List element",ii,"is conjugate to",order12repwordlist[jj],"using",kkwordlist[i1] cat ["B"] cat kkwordlist[i2]; 
//      end if;
//      i2+:=1;
//     end while;
//     i1+:=1;
//    end while;
//    jj+:=1;
//   end while;
//  end if;
//  if (conjugatefound eq 0) then
//   target:=order12list[ii];
//   jj:=1;
//   while ((jj le #order12replist) and (conjugatefound eq 0)) do
//    i1:=1;
//    while ((i1 le 288) and (conjugatefound eq 0)) do
//     i2:=1;
//     while ((i2 le 288) and (conjugatefound eq 0)) do
//      conjmat:=kklist[i1]*B*kklist[i2];
//      testmat:=target*conjmat*order12replist[jj]*conjmat^-1;
//      if (testmat eq testmat[1][1]*ID3) then 
//       conjugatefound:=1;
//       print "List element",ii,"is conjugate to the inverse of",order12repwordlist[jj],"using",kkwordlist[i1] cat ["B"] cat kkwordlist[i2]; 
//      end if;
//      i2+:=1;
//     end while;
//     i1+:=1;
//    end while;
//    jj+:=1;
//   end while;
//  end if;
// 
//  if (conjugatefound eq 0) then
//   target:=order12list[ii]^-1;
//   jj:=1;
//   while ((jj le #order12replist) and (conjugatefound eq 0)) do
//    i1:=1;
//    while ((i1 le 288) and (conjugatefound eq 0)) do
//     i2:=1;
//     while ((i2 le 288) and (conjugatefound eq 0)) do
//      conjmat:=kklist[i1]*BUIB*kklist[i2];
//      testmat:=target*conjmat*order12replist[jj]*conjmat^-1;
//      if (testmat eq testmat[1][1]*ID3) then 
//       conjugatefound:=1;
//       print "List element",ii,"is conjugate to",order12repwordlist[jj],"using",kkwordlist[i1] cat ["B*UI*B"] cat kkwordlist[i2]; 
//      end if;
//      i2+:=1;
//     end while;
//     i1+:=1;
//    end while;
//    jj+:=1;
//   end while;
//  end if;
//  if (conjugatefound eq 0) then
//   target:=order12list[ii];
//   jj:=1;
//   while ((jj le #order12replist) and (conjugatefound eq 0)) do
//    i1:=1;
//    while ((i1 le 288) and (conjugatefound eq 0)) do
//     i2:=1;
//     while ((i2 le 288) and (conjugatefound eq 0)) do
//      conjmat:=kklist[i1]*BUIB*kklist[i2];
//      testmat:=target*conjmat*order12replist[jj]*conjmat^-1;
//      if (testmat eq testmat[1][1]*ID3) then 
//       conjugatefound:=1;
//       print "List element",ii,"is conjugate to the inverse of",order12repwordlist[jj],"using",kkwordlist[i1] cat ["B*UI*B"] cat kkwordlist[i2]; 
//      end if;
//      i2+:=1;
//     end while;
//     i1+:=1;
//    end while;
//    jj+:=1;
//   end while;
//  end if;
// 
//  if (conjugatefound eq 0) then print "********* Problem with element number",ii," **************"; end if;
// end for;
// 
// UnsetLogFile();
// 
// // Confirming representatives for elements of order 2
// // **************************************************
// // The number of elements in order2list is 8
// List element 1 is conjugate to [V*V] using []
// List element 2 is conjugate to [V*V] using [U]
// List element 3 is conjugate to [V*V] using [UI]
// List element 4 is conjugate to [J*J*J*J*J*J] using []
// List element 5 is conjugate to [V*V] using [U*VI*U]
// List element 6 is conjugate to [V*V] using [V*UI]
// List element 7 is conjugate to [V*V] using [VI*U]
// List element 8 is conjugate to [B*UI*B*UI] using []
// 
// // Confirming representatives for elements of order 3
// // **************************************************
// // The number of elements in order3list is 42
// List element 1 is conjugate to [U] using []
// List element 2 is conjugate to the inverse of [U] using []
// List element 3 is conjugate to [U] using [V]
// List element 4 is conjugate to [U] using [VI]
// List element 5 is conjugate to [U] using [V*V]
// List element 6 is conjugate to [U] using [VI*U*VI]
// List element 7 is conjugate to [U] using [UI*V]
// List element 8 is conjugate to [U] using [V*U*VI]
// List element 9 is conjugate to [J*J*J*J] using []
// List element 10 is conjugate to [U*J*J*J*J] using []
// List element 11 is conjugate to [U] using [V*V*U*VI]
// List element 12 is conjugate to [U*J*J*J*J] using [V]
// List element 13 is conjugate to [U*J*J*J*J] using [VI]
// List element 14 is conjugate to [U*J*J*J*J] using [V*V]
// List element 15 is conjugate to [U*J*J*J*J] using [VI*U*VI]
// List element 16 is conjugate to [U*J*J*J*J] using [UI*V]
// List element 17 is conjugate to [U*J*J*J*J] using [V*U*VI]
// List element 18 is conjugate to the inverse of [J*J*J*J] using []
// List element 19 is conjugate to the inverse of [U] using [V*V*U*VI]
// List element 20 is conjugate to [U*J*J*J*J] using [V*V*U*VI]
// List element 21 is conjugate to the inverse of [U] using [VI*U*VI]
// List element 22 is conjugate to the inverse of [U] using [V*U*VI]
// List element 23 is conjugate to the inverse of [U] using [UI*V]
// List element 24 is conjugate to the inverse of [U] using [V]
// List element 25 is conjugate to the inverse of [U] using [V*V]
// List element 26 is conjugate to the inverse of [U] using [VI]
// List element 27 is conjugate to [U] using [U*B]
// List element 28 is conjugate to [B*U*V] using [V]
// List element 29 is conjugate to [B*U*V] using []
// List element 30 is conjugate to [B*U*V] using [V*V]
// List element 31 is conjugate to [B*U*V] using [VI]
// List element 32 is conjugate to the inverse of [B*U*V] using [UI]
// List element 33 is conjugate to the inverse of [B*U*V] using [V*UI]
// List element 34 is conjugate to the inverse of [U] using [UI*B*VI]
// List element 35 is conjugate to the inverse of [B*U*V] using [U*V*J*J*J*J*J*J*J*J*J*J*J]
// List element 36 is conjugate to the inverse of [B*U*V] using [V*V*UI]
// List element 37 is conjugate to the inverse of [B*U*V] using [B*V]
// List element 38 is conjugate to [B*U*V] using [B*U*J]
// List element 39 is conjugate to [B*U*V] using [B]
// List element 40 is conjugate to [B*U*V] using [U*B]
// List element 41 is conjugate to the inverse of [B*U*V] using [UI*B*V]
// List element 42 is conjugate to the inverse of [B*U*V] using [U*B*V]
// 
// // Confirming representatives for elements of order 4
// // **************************************************
// // The number of elements in order4list is 42
// List element 1 is conjugate to [V] using []
// List element 2 is conjugate to the inverse of [V] using []
// List element 3 is conjugate to [V] using [U]
// List element 4 is conjugate to [V] using [UI]
// List element 5 is conjugate to [V] using [VI*U]
// List element 6 is conjugate to [V] using [V*UI]
// List element 7 is conjugate to [J*J*J] using []
// List element 8 is conjugate to [V*J*J*J] using []
// List element 9 is conjugate to [V] using [U*VI*U]
// List element 10 is conjugate to [V*V*J*J*J] using []
// List element 11 is conjugate to [V*V*J*J*J] using [U]
// List element 12 is conjugate to [V*V*J*J*J] using [UI]
// List element 13 is conjugate to [V*J*J*J] using [U]
// List element 14 is conjugate to [V*J*J*J] using [UI]
// List element 15 is conjugate to [V*J*J*J] using [VI*U]
// List element 16 is conjugate to [V*J*J*J] using [V*UI]
// List element 17 is conjugate to the inverse of [V*J*J*J] using [U*VI*U]
// List element 18 is conjugate to [V*J*J*J] using [U*VI*U]
// List element 19 is conjugate to the inverse of [V*J*J*J] using [V*UI]
// List element 20 is conjugate to the inverse of [V*J*J*J] using [VI*U]
// List element 21 is conjugate to the inverse of [V*J*J*J] using [UI]
// List element 22 is conjugate to the inverse of [V*J*J*J] using [U]
// List element 23 is conjugate to the inverse of [J*J*J] using []
// List element 24 is conjugate to the inverse of [V] using [U*VI*U]
// List element 25 is conjugate to the inverse of [V*J*J*J] using []
// List element 26 is conjugate to [V*V*J*J*J] using [U*VI*U]
// List element 27 is conjugate to [V*V*J*J*J] using [V*UI]
// List element 28 is conjugate to [V*V*J*J*J] using [VI*U]
// List element 29 is conjugate to the inverse of [V] using [V*UI]
// List element 30 is conjugate to the inverse of [V] using [VI*U]
// List element 31 is conjugate to the inverse of [V] using [UI]
// List element 32 is conjugate to the inverse of [V] using [U]
// List element 33 is conjugate to [B*UI] using []
// List element 34 is conjugate to [B*UI] using [U*B*U*V]
// List element 35 is conjugate to [B*UI] using [UI*B*VI]
// List element 36 is conjugate to [B*UI] using [VI*U*B*U*V]
// List element 37 is conjugate to [B*UI] using [V*UI*B*VI]
// List element 38 is conjugate to [B*UI] using [V]
// List element 39 is conjugate to [B*UI] using [V*V]
// List element 40 is conjugate to [B*UI] using [VI]
// List element 41 is conjugate to the inverse of [V] using [B*U]
// List element 42 is conjugate to [V] using [V*U*B*UI]
// 
// // Confirming representatives for elements of order 6
// // **************************************************
// // The number of elements in order6list is 56
// List element 1 is conjugate to [V*V*U*J] using []
// List element 2 is conjugate to [V*V*U*J] using [V]
// List element 3 is conjugate to [V*V*U*J] using [U]
// List element 4 is conjugate to [J*J] using []
// List element 5 is conjugate to [V*V*U*J*J*J*J*J] using [V*V*UI]
// List element 6 is conjugate to [V*V*U*J] using [VI]
// List element 7 is conjugate to [V*V*J*J] using []
// List element 8 is conjugate to [V*V*U*J*J*J*J*J] using [U*V]
// List element 9 is conjugate to [V*V*J*J] using [U]
// List element 10 is conjugate to [V*V*J*J] using [UI]
// List element 11 is conjugate to [V*V*U*J*J*J*J*J] using [V*UI]
// List element 12 is conjugate to [V*V*U*J*J*J*J*J] using [UI]
// List element 13 is conjugate to the inverse of [V*V*J*J] using [U*VI*U]
// List element 14 is conjugate to the inverse of [V*V*J*J] using [V*UI]
// List element 15 is conjugate to the inverse of [V*V*J*J] using [VI*U]
// List element 16 is conjugate to [V*V*U*J*J*J*J*J] using []
// List element 17 is conjugate to [V*V*U*J*J*J*J*J] using [V]
// List element 18 is conjugate to [V*V*U*J*J*J*J*J] using [U]
// List element 19 is conjugate to the inverse of [V*V*U*J*J*J*J*J] using [VI]
// List element 20 is conjugate to [V*V*U*J*J*J*J*J] using [VI]
// List element 21 is conjugate to the inverse of [V*V*U*J*J*J*J*J] using []
// List element 22 is conjugate to the inverse of [V*V*U*J*J*J*J*J] using [U]
// List element 23 is conjugate to the inverse of [V*V*U*J*J*J*J*J] using [V]
// List element 24 is conjugate to [V*V*J*J] using [U*VI*U]
// List element 25 is conjugate to [V*V*J*J] using [V*UI]
// List element 26 is conjugate to [V*V*J*J] using [VI*U]
// List element 27 is conjugate to the inverse of [V*V*U*J*J*J*J*J] using [U*V]
// List element 28 is conjugate to the inverse of [V*V*U*J*J*J*J*J] using [UI]
// List element 29 is conjugate to the inverse of [V*V*U*J*J*J*J*J] using [V*UI]
// List element 30 is conjugate to the inverse of [J*J] using []
// List element 31 is conjugate to [V*V*U*J] using [V*V*UI]
// List element 32 is conjugate to the inverse of [V*V*U*J*J*J*J*J] using [V*V*UI]
// List element 33 is conjugate to the inverse of [V*V*J*J] using []
// List element 34 is conjugate to [V*V*U*J] using [U*V]
// List element 35 is conjugate to the inverse of [V*V*J*J] using [U]
// List element 36 is conjugate to the inverse of [V*V*J*J] using [UI]
// List element 37 is conjugate to [V*V*U*J] using [V*UI]
// List element 38 is conjugate to [V*V*U*J] using [UI]
// List element 39 is conjugate to [B*V*V*UI*J] using [V]
// List element 40 is conjugate to [B*V*V] using []
// List element 41 is conjugate to [B*V*V*UI*J] using [V*V]
// List element 42 is conjugate to the inverse of [B*V*V*UI*J] using [UI]
// List element 43 is conjugate to the inverse of [B*V*V*UI*J] using [U*V*J*J*J*J*J*J*J*J*J*J*J]
// List element 44 is conjugate to [B*V*V*UI*J] using []
// List element 45 is conjugate to [B*V*V*UI*J] using [VI]
// List element 46 is conjugate to the inverse of [B*V*V*UI*J] using [V*UI]
// List element 47 is conjugate to the inverse of [B*V*V*UI*J] using [V*V*UI]
// List element 48 is conjugate to the inverse of [B*V*V] using [J*J*J*J*J*J*J*J*J*J*J]
// List element 49 is conjugate to the inverse of [B*V*V*UI*J] using [B*V]
// List element 50 is conjugate to [B*V*V*UI*J] using [B*V]
// List element 51 is conjugate to [V*V*U*J] using [B*U*V]
// List element 52 is conjugate to [V*V*U*J] using [U*B*U*V]
// List element 53 is conjugate to [V*V*U*J] using [UI*B*U*V]
// List element 54 is conjugate to [V*V*U*J] using [U*J*B*V]
// List element 55 is conjugate to [V*V*U*J] using [V*U*B*U]
// List element 56 is conjugate to [V*V*U*J] using [J*B*V]
// 
// // Confirming representatives for elements of order 8
// // **************************************************
// // The number of elements in order8list is 42
// List element 1 is conjugate to [U*V*J] using [V*U]
// List element 2 is conjugate to [U*V*J] using [V]
// List element 3 is conjugate to [U*V*J] using []
// List element 4 is conjugate to [U*V*J] using [V*V]
// List element 5 is conjugate to [U*V*J] using [VI]
// List element 6 is conjugate to [U*V*J] using [U]
// List element 7 is conjugate to the inverse of [U*V*J] using [VI*U]
// List element 8 is conjugate to the inverse of [U*V*J] using [VI*U*VI]
// List element 9 is conjugate to the inverse of [U*V*J] using [V*V*U*VI]
// List element 10 is conjugate to the inverse of [U*V*J] using [U*VI]
// List element 11 is conjugate to the inverse of [U*V*J] using [V*U*VI]
// List element 12 is conjugate to the inverse of [U*V*J] using [V*V*U]
// List element 13 is conjugate to [U*V*J] using [VI*U]
// List element 14 is conjugate to [U*V*J] using [VI*U*VI]
// List element 15 is conjugate to [U*V*J] using [V*V*U*VI]
// List element 16 is conjugate to [U*V*J] using [U*VI]
// List element 17 is conjugate to [U*V*J] using [V*U*VI]
// List element 18 is conjugate to [U*V*J] using [V*V*U]
// List element 19 is conjugate to the inverse of [U*V*J] using [V*U]
// List element 20 is conjugate to the inverse of [U*V*J] using [V]
// List element 21 is conjugate to the inverse of [U*V*J] using []
// List element 22 is conjugate to the inverse of [U*V*J] using [V*V]
// List element 23 is conjugate to the inverse of [U*V*J] using [VI]
// List element 24 is conjugate to the inverse of [U*V*J] using [U]
// List element 25 is conjugate to [B*J] using []
// List element 26 is conjugate to [(B*J)^3] using []
// List element 27 is conjugate to the inverse of [B*J] using []
// List element 28 is conjugate to the inverse of [(B*J)^3] using []
// List element 29 is conjugate to the inverse of [B*J] using [UI*B*VI*U*J]
// List element 30 is conjugate to the inverse of [B*J] using [V*UI*B*VI*U*J]
// List element 31 is conjugate to the inverse of [B*J] using [J*J*J*J*J*J*J*J*J*J*J*B*U]
// List element 32 is conjugate to the inverse of [B*J] using [V*V*UI*B*VI*U*J]
// List element 33 is conjugate to [B*J] using [V*U*B*V*U]
// List element 34 is conjugate to [B*J] using [U*B*V*U]
// List element 35 is conjugate to [B*J] using [V*V*U*B*V*U]
// List element 36 is conjugate to [B*J] using [VI*U*B*V*U]
// List element 37 is conjugate to the inverse of [U*V*J] using [B*V]
// List element 38 is conjugate to the inverse of [U*V*J] using [U*B*V]
// List element 39 is conjugate to the inverse of [U*V*J] using [UI*B*V]
// List element 40 is conjugate to [U*V*J] using [U*J*B*VI]
// List element 41 is conjugate to [U*V*J] using [V*U*B]
// List element 42 is conjugate to [U*V*J] using [J*B*VI]
// 
// // Confirming representatives for elements of order 24
// // ***************************************************
// // The number of elements in order24list is 62
// List element 1 is conjugate to [U*V] using [V]
// List element 2 is conjugate to [U*V] using []
// List element 3 is conjugate to the inverse of [U*V] using [U]
// List element 4 is conjugate to [U*V] using [V*V]
// List element 5 is conjugate to [U*V] using [VI]
// List element 6 is conjugate to [U*V] using [U]
// List element 7 is conjugate to [V*U*J*J] using [V*UI]
// List element 8 is conjugate to [V*U*J*J] using []
// List element 9 is conjugate to [V*U*J*J] using [VI]
// List element 10 is conjugate to [V*U*J*J] using [V]
// List element 11 is conjugate to [V*U*J*J] using [V*V]
// List element 12 is conjugate to the inverse of [V*U*J*J] using [U*V]
// List element 13 is conjugate to the inverse of [V*U*J*J] using [VI*U*V*V]
// List element 14 is conjugate to the inverse of [V*U*J*J] using [V*UI*V]
// List element 15 is conjugate to [V*U*J*J] using [UI]
// List element 16 is conjugate to the inverse of [V*U*J*J] using [U*V*V]
// List element 17 is conjugate to the inverse of [V*U*J*J] using [UI*V]
// List element 18 is conjugate to the inverse of [V*U*J*J] using [V*V*UI]
// List element 19 is conjugate to the inverse of [U*V] using [VI*U]
// List element 20 is conjugate to the inverse of [U*V] using [VI*U*VI]
// List element 21 is conjugate to the inverse of [U*V] using [V*V*U*VI]
// List element 22 is conjugate to the inverse of [U*V] using [U*VI]
// List element 23 is conjugate to the inverse of [U*V] using [V*U*VI]
// List element 24 is conjugate to [U*V] using [VI*U]
// List element 25 is conjugate to [U*V] using [VI*U*VI]
// List element 26 is conjugate to [U*V] using [V*V*U*VI]
// List element 27 is conjugate to the inverse of [U*V] using [V*V*U]
// List element 28 is conjugate to [U*V] using [U*VI]
// List element 29 is conjugate to [U*V] using [V*U*VI]
// List element 30 is conjugate to [U*V] using [V*V*U]
// List element 31 is conjugate to [V*U*J*J] using [U*V]
// List element 32 is conjugate to [V*U*J*J] using [VI*U*V*V]
// List element 33 is conjugate to [V*U*J*J] using [V*UI*V]
// List element 34 is conjugate to [V*U*J*J] using [U*V*V]
// List element 35 is conjugate to [V*U*J*J] using [UI*V]
// List element 36 is conjugate to the inverse of [V*U*J*J] using [V*UI]
// List element 37 is conjugate to the inverse of [V*U*J*J] using []
// List element 38 is conjugate to the inverse of [V*U*J*J] using [VI]
// List element 39 is conjugate to [V*U*J*J] using [V*V*UI]
// List element 40 is conjugate to the inverse of [V*U*J*J] using [V]
// List element 41 is conjugate to the inverse of [V*U*J*J] using [V*V]
// List element 42 is conjugate to the inverse of [V*U*J*J] using [UI]
// List element 43 is conjugate to the inverse of [U*V] using [V*U]
// List element 44 is conjugate to the inverse of [U*V] using [V]
// List element 45 is conjugate to the inverse of [U*V] using []
// List element 46 is conjugate to the inverse of [U*V] using [V*V]
// List element 47 is conjugate to the inverse of [U*V] using [VI]
// List element 48 is conjugate to [U*V] using [V*U]
// List element 49 is conjugate to the inverse of [U*V] using [V*UI*B]
// List element 50 is conjugate to the inverse of [U*V] using [V*V*UI*B]
// List element 51 is conjugate to the inverse of [U*V] using [U*J*J*J*J*J*J*J*J*J*J*J*B*V]
// List element 52 is conjugate to the inverse of [U*V] using [UI*B]
// List element 53 is conjugate to [U*V] using [U*B]
// List element 54 is conjugate to [U*V] using [V*U*B]
// List element 55 is conjugate to [U*V] using [VI*U*B]
// List element 56 is conjugate to [U*V] using [V*V*U*
// List element 57 is conjugate to [U*V] using [V*B*UI*B]
// List element 58 is conjugate to the inverse of [U*V] using [V*J*J*J*J*J*J*J*J*J*J*J*B*UI*B*U]
// List element 59 is conjugate to the inverse of [U*V] using [VI*B*UI*B]
// List element 60 is conjugate to [U*V] using [U*V*B*UI*B]
// List element 61 is conjugate to [U*V] using [UI*V*B*UI*B]
// List element 62 is conjugate to the inverse of [U*V] using [U*VI*B*UI*B]
// 
// // Confirming representatives for elements of order 12
// // ***************************************************
// // The number of elements in order12list is 156
// List element 1 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [V*V ]
// List element 2 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [V ]
// List element 3 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [V ]
// List element 4 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [V*V ]
// List element 5 is conjugate to [U*V*V] using [UI ]
// List element 6 is conjugate to the inverse of [U*V*V] using []
// List element 7 is conjugate to the inverse of [U*V*V] using [VI ]
// List element 8 is conjugate to [U*V*V] using [VI ]
// List element 9 is conjugate to [U*V*V] using []
// List element 10 is conjugate to [J] using []
// List element 11 is conjugate to [U*VI*J*J] using [U ]
// List element 12 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [UI*V ]
// List element 13 is conjugate to [U*J] using []
// List element 14 is conjugate to [U*V*V] using [V ]
// List element 15 is conjugate to [V*V*J] using []
// List element 16 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using []
// List element 17 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [VI ]
// List element 18 is conjugate to [U*J] using [V ]
// List element 19 is conjugate to [V*V*J] using [U ]
// List element 20 is conjugate to [V*V*J] using [UI ]
// List element 21 is conjugate to [U*VI*J*J*J] using [U ]
// List element 22 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [U ]
// List element 23 is conjugate to [U*VI*J*J] using [VI ]
// List element 24 is conjugate to [U*VI*J*J] using []
// List element 25 is conjugate to the inverse of [U*V*V] using [VI*U ]
// List element 26 is conjugate to [U*J] using [VI ]
// List element 27 is conjugate to [U*J] using [V*V ]
// List element 28 is conjugate to the inverse of [U*V*V] using [U ]
// List element 29 is conjugate to the inverse of [U*V*V] using [V*U ]
// List element 30 is conjugate to [U*J*J*J] using []
// List element 31 is conjugate to the inverse of [U*V*V] using [V*V*U ]
// List element 32 is conjugate to [U*VI*J*J] using [V*V ]
// List element 33 is conjugate to [U*VI*J*J*J] using [VI ]
// List element 34 is conjugate to [U*VI*J*J] using [V ]
// List element 35 is conjugate to [U*VI*J*J*J] using []
// List element 36 is conjugate to [U*J*J*J] using [V ]
// List element 37 is conjugate to the inverse of [U*VI*J*J*J*J*J*J] using [UI*V ]
// List element 38 is conjugate to [U*VI*J*J] using [UI*V ]
// List element 39 is conjugate to [U*VI*J*J*J] using [V*V ]
// List element 40 is conjugate to [U*VI*J*J*J] using [V ]
// List element 41 is conjugate to [U*J] using [VI*U*VI ]
// List element 42 is conjugate to [U*J*J*J] using [VI ]
// List element 43 is conjugate to [U*J*J*J] using [V*V ]
// List element 44 is conjugate to [U*J] using [UI*V ]
// List element 45 is conjugate to [U*J] using [V*U*VI ]
// List element 46 is conjugate to [J*J*J*J*J] using []
// List element 47 is conjugate to [U*VI*J*J*J*J*J*J] using [U ]
// List element 48 is conjugate to [U*VI*J*J*J] using [UI*V ]
// List element 49 is conjugate to the inverse of [U*J*J*J] using [V*V*U*VI ]
// List element 50 is conjugate to [U*J] using [V*V*U*VI ]
// List element 51 is conjugate to the inverse of [V*V*J] using [U*VI*U ]
// List element 52 is conjugate to the inverse of [U*VI*J*J*J*J*J*J] using [V ]
// List element 53 is conjugate to the inverse of [U*VI*J*J*J*J*J*J] using [V*V ]
// List element 54 is conjugate to the inverse of [U*J*J*J] using [VI*U*VI ]
// List element 55 is conjugate to the inverse of [V*V*J] using [V*UI ]
// List element 56 is conjugate to the inverse of [V*V*J] using [VI*U ]
// List element 57 is conjugate to the inverse of [U*VI*J*J*J*J*J*J] using []
// List element 58 is conjugate to [U*VI*J*J*J*J*J*J] using [VI ]
// List element 59 is conjugate to the inverse of [U*VI*J*J*J*J*J*J] using [VI ]
// List element 60 is conjugate to [U*VI*J*J*J*J*J*J] using []
// List element 61 is conjugate to [U*J*J*J] using [VI*U*VI ]
// List element 62 is conjugate to the inverse of [U*J*J*J] using [V*U*VI ]
// List element 63 is conjugate to the inverse of [U*J*J*J] using [UI*V ]
// List element 64 is conjugate to [U*J*J*J] using [UI*V ]
// List element 65 is conjugate to [U*J*J*J] using [V*U*VI ]
// List element 66 is conjugate to the inverse of [J*J*J*J*J] using []
// List element 67 is conjugate to the inverse of [U*VI*J*J*J] using [UI*V ]
// List element 68 is conjugate to the inverse of [U*VI*J*J*J*J*J*J] using [U ]
// List element 69 is conjugate to the inverse of [U*J] using [V*V*U*VI ]
// List element 70 is conjugate to [U*J*J*J] using [V*V*U*VI ]
// List element 71 is conjugate to [V*V*J] using [U*VI*U ]
// List element 72 is conjugate to [U*VI*J*J*J*J*J*J] using [V*V ]
// List element 73 is conjugate to [U*VI*J*J*J*J*J*J] using [V ]
// List element 74 is conjugate to the inverse of [U*J] using [VI*U*VI ]
// List element 75 is conjugate to [V*V*J] using [V*UI ]
// List element 76 is conjugate to [V*V*J] using [VI*U ]
// List element 77 is conjugate to the inverse of [U*VI*J*J] using [UI*V ]
// List element 78 is conjugate to [U*VI*J*J*J*J*J*J] using [UI*V ]
// List element 79 is conjugate to the inverse of [U*VI*J*J*J] using [V ]
// List element 80 is conjugate to the inverse of [U*VI*J*J*J] using [V*V ]
// List element 81 is conjugate to the inverse of [U*J*J*J] using [V ]
// List element 82 is conjugate to the inverse of [U*J] using [V*U*VI ]
// List element 83 is conjugate to the inverse of [U*J] using [UI*V ]
// List element 84 is conjugate to the inverse of [U*J*J*J] using [V*V ]
// List element 85 is conjugate to the inverse of [U*J*J*J] using [VI ]
// List element 86 is conjugate to [U*V*V] using [V*V*U ]
// List element 87 is conjugate to the inverse of [U*J*J*J] using []
// List element 88 is conjugate to the inverse of [U*VI*J*J*J] using []
// List element 89 is conjugate to the inverse of [U*VI*J*J] using [V ]
// List element 90 is conjugate to the inverse of [U*VI*J*J*J] using [VI ]
// List element 91 is conjugate to the inverse of [U*VI*J*J] using [V*V ]
// List element 92 is conjugate to [U*V*V] using [VI*U ]
// List element 93 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [U ]
// List element 94 is conjugate to the inverse of [U*VI*J*J*J] using [U ]
// List element 95 is conjugate to the inverse of [U*VI*J*J] using []
// List element 96 is conjugate to the inverse of [U*VI*J*J] using [VI ]
// List element 97 is conjugate to the inverse of [U*J] using [V ]
// List element 98 is conjugate to [U*V*V] using [V*U ]
// List element 99 is conjugate to [U*V*V] using [U ]
// List element 100 is conjugate to the inverse of [U*J] using [V*V ]
// List element 101 is conjugate to the inverse of [U*J] using [VI ]
// List element 102 is conjugate to the inverse of [J] using []
// List element 103 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [UI*V ]
// List element 104 is conjugate to the inverse of [U*VI*J*J] using [U ]
// List element 105 is conjugate to the inverse of [U*V*V] using [V ]
// List element 106 is conjugate to the inverse of [U*J] using []
// List element 107 is conjugate to the inverse of [V*V*J] using []
// List element 108 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [VI ]
// List element 109 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using []
// List element 110 is conjugate to the inverse of [U*V*V] using [UI ]
// List element 111 is conjugate to the inverse of [V*V*J] using [U ]
// List element 112 is conjugate to the inverse of [V*V*J] using [UI ]
// List element 113 is conjugate to [B*V] using []
// List element 114 is conjugate to [(B*V)^(-5)] using []
// List element 115 is conjugate to [U*V*V] using [U*B ]
// List element 116 is conjugate to [U*V*V] using [V*U*B ]
// List element 117 is conjugate to the inverse of [U*V*V] using [UI*B*V ]
// List element 118 is conjugate to the inverse of [(B*V)^(-5)] using [J*J*J*J*J*J*J*J*J*J*J ]
// List element 119 is conjugate to the inverse of [B*V] using [J*J*J*J*J*J*J*J*J*J*J ]
// List element 120 is conjugate to the inverse of [V*V*J] using [J*J*J*J*J*J*J*J*J*J*J*B ]
// List element 121 is conjugate to the inverse of [U*VI*J*J] using [J*J*J*J*J*J*J*J*J*J*J*B*U ]
// List element 122 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [J*J*J*J*J*J*J*J*J*J*J*B*UI*V ]
// List element 123 is conjugate to the inverse of [J] using [J*J*J*J*J*J*J*J*J*J*J*B ]
// List element 124 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [U*J*J*J*J*J*J*J*J*J*J*J*B*V ]
// List element 125 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [UI*B ]
// List element 126 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [V*UI*B ]
// List element 127 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [V*V*UI*B ]
// List element 128 is conjugate to the inverse of [U*V*V] using [U*J*J*J*J*J*J*J*J*J*J*J*B*UI ]
// List element 129 is conjugate to [U*V*V] using [V*V*U*B ]
// List element 130 is conjugate to [U*V*V] using [VI*U*B ]
// List element 131 is conjugate to the inverse of [U*V*V] using [V*V*UI*B*V ]
// List element 132 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [V*V*U*B*V*V ]
// List element 133 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [VI*U*B*V*V ]
// List element 134 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [J*B*UI*V ]
// List element 135 is conjugate to [J] using [J*B ]
// List element 136 is conjugate to [V*V*J] using [J*B ]
// List element 137 is conjugate to [U*VI*J*J] using [J*B*U ]
// List element 138 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [V*U*B*V*V ]
// List element 139 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [U*B*V*V ]
// List element 140 is conjugate to the inverse of [U*V*V] using [V*UI*B*V ]
// List element 141 is conjugate to the inverse of [U*J] using [UI*B ]
// List element 142 is conjugate to the inverse of [U*J] using [U*B ]
// List element 143 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [V*B*UI*B*UI*V ]
// List element 144 is conjugate to [U*J] using [U*J*B*VI ]
// List element 145 is conjugate to the inverse of [U*J] using [B ]
// List element 146 is conjugate to [(B*V)^(-5)] using [B*U ]
// List element 147 is conjugate to the inverse of [B*V] using [B*U ]
// List element 148 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [V*J*J*J*J*J*J*J*J*J*J*J*B*UI*B*V*V ]
// List element 149 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [VI*B*UI*B*V ]
// List element 150 is conjugate to the inverse of [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [U*VI*B*UI*B*V ]
// List element 151 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [U*V*B*UI*B*UI*V ]
// List element 152 is conjugate to [U*VI*J*J*J*J*J*J*J*J*J*J*J] using [UI*V*B*UI*B*UI*V ]
// List element 153 is conjugate to the inverse of [(B*V)^(-5)] using [V*U*B*V*U ]
// List element 154 is conjugate to [B*V] using [V*U*B*V*U ]
// List element 155 is conjugate to [U*J] using [V*U*B ]
// List element 156 is conjugate to [U*J] using [J*B*VI ]

// End of calculations for Proposition 1.
// **************************************

// Calculations for the table of representatives of order d, for the various d's.
// ******************************************************************************

// We shall shortly be listing the 24 elements of $\bar\Gamma$ which fix 
// $P=(c(\zeta-1)/\sqrt{r-1},(\zeta-1)/\sqrt{r-1})$. To find
// this list, note that if $g.P=P$, then $d(g.O,O)\le d(g.O,g.P)+d(g.P,P)+d(P,O)=2d(P,0)$,
// since $\bar\Gamma$ acts on $B^2_\C$ preserving the hyperboli metric $d$. From formula (3.3)
// in [CS2], we have $\cosh(2d(P,0))=2\cosh^2(d(P,0))-1$=2/(1-|P|^2)-1=(1+|P|^2)/(1-|P|^2)$,
// where $|z|=\sqrt{|z_1|^2+|z_2|^2}$ for $z=(z_1,z_2)\in B^2_\C$.
// We calculate that $|P|^2=2r-3$, where $r=\sqrt3$, and so $\cosh(2d(P,0))=r+1$. So
// $g.P=P$ implies that $d(g.0,0)\le\cosh^{-1}(r+1)$. This means that
// $\cosh^2(d(g.0,0))\le(r+1)^2=2r+4$, and so $g\in K \cup KbK \cup Kbu^{-1}bK$ (see
// the remark after Proposition 3.2 in [CS2]). So to find the $g\in\bar\Gamma$ such
// that $g.P=P$, we need only search amongst the elements in the finite set 
// $S=K\cup KbK \cup Kbu^{-1}bK$ to find those satisfying $g.P=P$.

// As explained in Section 1.1, if $g$ is a matrix unitary with respect to the diagonal
// form $F$, then the action of $gZ$ on $B^2_\C^2$ is obtained by conjugating $g$ by
// the diagonal matrix $D$ with diagonal matrix with diagonal entries 1, 1 and $\sqrt{r-1}$,
// and $(gZ).(z,w)=(z',w')$ means that $(DgD^{-1})(z,w,1)^t=\lambda(z',w',1)^t$ for some
// scalar $\lambda$. So $(gZ).(z,w)=(z,w)$ if and only if $(z,w,1/\sqrt{r-1})$ is an
// eigenvalue of $g$. That is, $(gZ)(z'/\sqrt{r-1},w'/\sqrt{r-1})$ if and only if
// $(z',w',1)$ is an eigenvector of $g$.

// So the point $P=(c(\zeta-1)/\sqrt{r-1},(\zeta-1)/\sqrt{r-1})$ is fixed by $gZ\in\bar\Gamma$
// if and only if $(c(\zeta-1),\zeta-1,1)^t$ is an eigenvector of $g$. We set up this
// vector as follows:

VecSpaceL:=RMatrixSpace(L,3,1);
Orig:=VecSpaceL![0,0,1];
c:=(r-1)*(z^3-1)/2;
Pvec:=VecSpaceL![c*(z-1),z-1,1];

// Her is a list of the 24 elements fixing $P$. It is a subset of the above set $S$,
// but it is also equal to $\langle u,b\rangle$. The following lists the 24 elements
// making it evident that they are in $K\cup KbK \cup Kbu^{-1}bK$. It clearly
// contains $\langle b,u\rangle$, and for the eight elements not obviously in $\langle b,u\rangle$,
// a second formula for the group element is given making this evident.

Pfixerlist:=[MatsL |
ID3,
U,
UI,
B,
B*U,
B*UI,
U*B,
U*B*U,
U*B*UI,
UI*B,
UI*B*U,
UI*B*UI,
V*U*B*J,     // = UI*B^2,
V*U*B*U*J,   // = UI*B^2*U,
V*U*B*UI*J,  // = UI*B^2*UI,
J*B*U*V,     // = B^2*UI,
J*B*VI*J,    // = B^2,
J*B*VI*U*J,  // = B^2*U,
U*J*B*U*V,   // = U*B^2*UI,
U*J*B*VI*J,  // = U*B^2,
U*J*B*VI*U*J,// = U*B^2*U
B*UI*B,
B*UI*B*U,
B*UI*B*UI];

for ii in [1 .. #Pfixerlist] do
 tmpvec:=Pfixerlist[ii]*Pvec;
 if (tmpvec ne tmpvec[3][1]*Pvec) then print "element number",ii," of Pfixerlist does not fix Pvec."; end if;
end for;
print "finished checking that the elements of Pfixerlist fix Pvec.";

// We now check a presentation for the stabilizer of $P$,
// and check that $P$ belongs to 4 distinct mirrors of type $A$.
fmat2:=B*UI;
fmat3:=J*B*VI*J;
fmat3d:=U;
fmatz:=(B*UI)^2;

r1:=ID3;
r2:=fmat2;
r3:=fmat3;
r4:=fmat3d*fmat2;
// Checking that Pvec is in r_nu(M_c).
// The following are zero:
tmpvec:=r1^-1*Pvec; tmpvec[1][1]-c*tmpvec[2][1];
tmpvec:=r2^-1*Pvec; tmpvec[1][1]-c*tmpvec[2][1];
tmpvec:=r3^-1*Pvec; tmpvec[1][1]-c*tmpvec[2][1];
tmpvec:=r4^-1*Pvec; tmpvec[1][1]-c*tmpvec[2][1];
// Checking that the r_nu(M_c) are distinct. 
// The following are NOT zero:
print "The following six numbers are not zero:";
MM:=r1^-1*r2;MM[1][2]-c*(c*MM[2][1]-MM[1][1]+MM[2][2]);
MM:=r1^-1*r3;MM[1][2]-c*(c*MM[2][1]-MM[1][1]+MM[2][2]);
MM:=r1^-1*r4;MM[1][2]-c*(c*MM[2][1]-MM[1][1]+MM[2][2]);
MM:=r2^-1*r3;MM[1][2]-c*(c*MM[2][1]-MM[1][1]+MM[2][2]);
MM:=r2^-1*r4;MM[1][2]-c*(c*MM[2][1]-MM[1][1]+MM[2][2]);
MM:=r3^-1*r4;MM[1][2]-c*(c*MM[2][1]-MM[1][1]+MM[2][2]);
print "Finished checking that the r_nu(M_c) are distinct.";

// The following are zero:
fmatz*Pvec - Pvec;
fmat2*Pvec - Pvec;
fmat3*Pvec - (1/z)*Pvec;
fmat3d*Pvec-Pvec;
print "Finished checking that fmatz, fmat2, fmat3 and fmat3d fix P.";

// The following are zero:
fmat2^2 - fmatz;
fmat3^3 - (1/z^3)*ID3;
fmat3d^3 - ID3;
fmatz^2 - ID3;
fmat3d*fmat3*fmat2 - (1/z)*ID3;
fmat2*fmatz-fmatz*fmat2;
fmat3*fmatz-fmatz*fmat3;
fmat3d*fmatz-fmatz*fmat3d;
print "Finished checking the relations in the stabilizer of P.";

// Calculations concerning the fixed points of $bv$, $buv$ and $bj$.
// *****************************************************************

// We now list three elements $\xi$ of $B(\C^2)$ for which 
// $\bar\Gamma_\xi$ is cyclic of order 12, 8 and 3, respectively.
// For the last two of these, the entries of $\xi$ are in an
// extension file of $\ell$.

// The following vector vec12 corresponds to the point $\xi_{12}=(0,(\zeta-1)/\sqrt{r-1})$ of $B^2_\C$:
// We have $|\xi_{12}|^2=(r-1)/2$. This is smaller than $|P|^2$, and so to find
// the $gZ\in\bar\Gamma$ which fix~$\xi_{12}$, we need only look amongst the
// $gZ$ in the set $S=K \cup KbK \cup Kbu^{-1}bK$, seeking those having
// vec12 as eigenvector.

vec12:=VecSpaceL![0,z-1,1];
gamma12:=B*V; // This is an element of \bar\Gamma of order 12
// the following is zero:
gamma12*vec12-vec12;

// The following vector vec8 corresponds to the point $\xi_8=(0,(1-2\sin(\pi/12))\zeta^3/\sqrt{r-1})$ of $B^2_\C$:
// We have $|\xi_8|^2=r-\sqrt{2}$. This is smaller than $|P|^2$, and so to find
// the $gZ\in\bar\Gamma$ which fix~$\xi_8$, we need only look amongst the
// $gZ$ in the set $S=K \cup KbK \cup Kbu^{-1}bK$, seeking those having
// vec8 as eigenvector.

P8<x8>:=PolynomialRing(L);
L8<lam8>:=ext<L | x8^2+z^3>;
VecSpaceL8:=RMatrixSpace(L8,3,1);
vec8:=VecSpaceL8![0,(lam8*(z-1)*(z^2-1)+1)*z^3,1];
// The middle entry of vec8 is equal to $(1-2\sin(\pi/12))\zeta^3$ when $\lambda=e^{-i\pi/4}$.
MatsL8:=MatrixRing(L8,3);
gamma8:=MatsL8!(z^-1*B*J); // This is an element of \bar\Gamma of order 8
// the following is zero:
gamma8*vec8-lam8*vec8;

// The following vector vec3 corresponds to the point $Q$ of $B^2_\C$:
// We have $|Q|^2=0.381315...$. This is smaller than $|P|^2$, and so to find
// the $gZ\in\bar\Gamma$ which fix~$\xi_3$, we need only look amongst the
// $gZ$ in the set $S=K \cup KbK \cup Kbu^{-1}bK$, seeking those having
// vec3 as eigenvector.

P3<x3>:=PolynomialRing(L);
L3<lam3>:=ext<L | x3^3-z^(-1)>;
VecSpaceL3:=RMatrixSpace(L3,3,1);
MatsL3:=MatrixRing(L3,3);
gamma3:=MatsL3!(B*U*V);  // This is an element of \bar\Gamma of order 3

vec3:=VecSpaceL3![(z^3-z^2-z+1)+(z^2-z+1)*lam3+(-z^3+z^2-1)*lam3^2,z^3-(z-1)*lam3^2,1];
// the following is zero:
gamma3*vec3-lam3*vec3;


// Calculations for Lemma 2:
// *************************

// Define the following elements of $K$:
kpmm:=ID3;
kmmp:=V;
kmmm:=V^2;
kpmp:=V^3;
kmpp:=UI*V^2*U;
kmpm:=V*UI*V^2*U;  
kppp:=V^2*UI*V^2*U;
kppm:=V^3*UI*V^2*U;

// The condition on a matrix $(k_{ij})$ that for $k(M_c)=M_\alpha$
// to hold is that $k_{13}=\alpha k_{23}$ and that $ck_{11}+k_{12}=\alpha(ck_{21}+k_{22})$.
// So we check this:

cppp:=(r+1)*(z^3+1)/2;
cppm:=(r+1)*(z^3-1)/2;
cpmp:=(r-1)*(z^3+1)/2;
cpmm:=(r-1)*(z^3-1)/2;

cmpp:=-(r+1)*(z^3+1)/2;
cmpm:=-(r+1)*(z^3-1)/2;
cmmp:=-(r-1)*(z^3+1)/2;
cmmm:=-(r-1)*(z^3-1)/2;

// // The following are zero:
// kppp[1][3]-cppp*kppp[2][3];
// c*kppp[1][1]+kppp[1][2]-cppp*(c*kppp[2][1]+kppp[2][2]);
// kppm[1][3]-cppm*kppm[2][3];
// c*kppm[1][1]+kppm[1][2]-cppm*(c*kppm[2][1]+kppm[2][2]);
// kpmp[1][3]-cpmp*kpmp[2][3];
// c*kpmp[1][1]+kpmp[1][2]-cpmp*(c*kpmp[2][1]+kpmp[2][2]);
// kpmm[1][3]-cpmm*kpmm[2][3];
// c*kpmm[1][1]+kpmm[1][2]-cpmm*(c*kpmm[2][1]+kpmm[2][2]);
// 
// kmpp[1][3]-cmpp*kmpp[2][3];
// c*kmpp[1][1]+kmpp[1][2]-cmpp*(c*kmpp[2][1]+kmpp[2][2]);
// kmpm[1][3]-cmpm*kmpm[2][3];
// c*kmpm[1][1]+kmpm[1][2]-cmpm*(c*kmpm[2][1]+kmpm[2][2]);
// kmmp[1][3]-cmmp*kmmp[2][3];
// c*kmmp[1][1]+kmmp[1][2]-cmmp*(c*kmmp[2][1]+kmmp[2][2]);
// kmmm[1][3]-cmmm*kmmm[2][3];
// c*kmmm[1][1]+kmmm[1][2]-cmmm*(c*kmmm[2][1]+kmmm[2][2]);

// We also check that $k(M_c)=M_c$ for $k=J$ and $k=U$.
// So the stabilizer $K_c$ of $M_c$ in $K$ is $\langle j,u\rangle$,
// which has order 36. Hence the orbit of $M_c$ has order $288/36=8$.
// // The following are zero:
// J[1][3]-c*J[2][3];
// c*J[1][1]+J[1][2]-c*(c*J[2][1]+J[2][2]);
// U[1][3]-c*U[2][3];
// c*U[1][1]+U[1][2]-c*(c*U[2][1]+U[2][2]);
// 
kzero:=ID3;
kplusi:=U*J;
kminus1:=V*U*J;
kminusi:=V^2*U*J;
kplus1:=V^3*U*J;
kinfty:=UI*V^2*U*J^6;
// 
// // For $\alpha\ne\infty$, the condition on a matrix $(k_{ij})$ that for $k(M_0)=M_\alpha$
// // to hold is that $k_{13}=\alpha k_{23}$ and that $k_{12}=\alpha k_{22}$.
// // The condition on a matrix $(k_{ij})$ that for $k(M_0)=M_\infty$
// // to hold is that $k_{23}=0=k_{22}$.
// // So we check this:
// alphazero:=0;
// alphaplus1:=1;
// alphaminus1:=-1;
// alphaplusi:=z^3;
// alphaminusi:=-z^3;
// 
// // The following are zero:
// kzero[1][3]-alphazero*kzero[2][3];
// kzero[1][2]-alphazero*kzero[2][2];
// 
// kplus1[1][3]-alphaplus1*kplus1[2][3];
// kplus1[1][2]-alphaplus1*kplus1[2][2];
// 
// kminus1[1][3]-alphaminus1*kminus1[2][3];
// kminus1[1][2]-alphaminus1*kminus1[2][2];
// 
// kplusi[1][3]-alphaplusi*kplusi[2][3];
// kplusi[1][2]-alphaplusi*kplusi[2][2];
// 
// kminusi[1][3]-alphaminusi*kminusi[2][3];
// kminusi[1][2]-alphaminusi*kminusi[2][2];
// 
// kinfty[2][3];
// kinfty[2][2];

// We also check that $k(M_0)=M_0$ for $k=J$ and $k=V$.
// So the stabilizer $K_c$ of $M_c$ in $K$ is $\langle j,v\rangle$,
// which has order 48. Hence the orbit of $M_c$ has order $288/48=6$.
// // The following are zero:
// J[1][3];
// J[1][2];
// V[1][3];
// V[1][2];

// End of calculations for Lemma 2.
// ********************************

// End of calculations for Section 1.3.
// ************************************

// Calculations for Theorem 2.
// ***************************

// Here are the generators of the subgroup $\Pi$ of $\bar\Gamma$:
A1:=V*U*VI*J^4*B*U*V*J^2;
A2:=V^2*U*B*U*VI*U*V^2*J;
A3:=UI*V^2*U*J^9*B*VI*U*VI*J^8;

A1I:=A1^-1;
A2I:=A2^-1;
A3I:=A3^-1;

// The assertions in Theorem 2 are checked in the file cs-surface-presentations_magma.txt

// Calculations about $j^4$ normalizing $\Pi$ (page 8):
// ****************************************************
// The following are zero:

J^4*A1*J^8 - z^3*A3*A2I^3*A3^3*A1;
J^4*A2*J^8 - (1/z)*A3I;
J^4*A3*J^8 - (1/z)*A1I*A2I*A1*A2^2*A1I*A2I*A1*A3I*A1I*A2*A1;

// Here are the relations for a presentation of $\Pi$.
// It was found using Magma's Rewrite routine. It turned out
// that Magma gave a simpler result if one applied the routine
// to Pi, conjugated by U*V^2*UI*J^3:
// A1D:=V^2*UI*JI*B*V*UI;
// A2D:=V*UI*JI^2*B*J^2;
// A3D:=VI*J^3*B*U*VI*JI;
// // The following are zero:
// A1D - (U*V^2*UI*J^3)*A1*(U*V^2*UI*J^3)^-1;
// A2D - (U*V^2*UI*J^3)*A2*(U*V^2*UI*J^3)^-1;
// A3D - (U*V^2*UI*J^3)*A3*(U*V^2*UI*J^3)^-1;

// The following are zero:
A1I*A3I*A1*A2^2*A1I*A2I*A3I^3*A2*A1I^2*A3I^3 - z^5*ID3;
A2^2*A1I*A2I*A1*A3^3*A1*A2I^3*A3^3*A1*A3*A1 -z^3*ID3;
A3^3*A1^2*A2*A1I*A2I*A1*A3*A2I^3*A3^3*A1*A2 - z^3*ID3;
A3I*A1I*A2I^2*A3^3*A1*A2I*A1I*A3I^3*A2^3*A3*A1 - ID3;
A2I^3*A3^3*A1*A2*A3I^3*A2^3*A3I*A1I*A2I*A1*A3*A1I - z^8*ID3;
A3I*A1I*A3I^3*A2^2*A1I^2*A3I^3*A2*A1*A3*A1I*A2*A1*A2I^2*A1I*A3I - z^9*ID3;
A3^3*A1^2*A2*A3I*A1I*A2I^2*A3^3*A1^2*A3^3*A1*A3I*A1I*A2I - (1/z)*ID3;
A2I^3*A3^3*A1^2*A2^2*A1I*A2I*A1*A3I*A1I^2*A3I^3*A2^2*A1*A3*A1I -(1/z^4)*ID3;
A3*A2I^3*A3^3*A1*A3^3*A1*A3^3*A1^2*A2I*A1I^2*A3I^3*A2^2*A1 - (1/z)*ID3;
A1I*A3I^3*A2^3*A1I*A2I*A1I*A3I^3*A2^4*A1*A2I*A1I^2*A3I^3*A1*A2I^2*A3^3*A1*A3I*A1I*A2I-z^5*ID3;
A3I^2*A2^4*A1*A2I*A1I^2*A3I^3*A1*A2I*A1I^2*A2*A1*A2I^2*A1I*A3*A1^2*A3*A1I*A2*A1*A2I^2*A1I*A2*A1I*A3I^3*A2*A1I*A3I - z*ID3;
A2I^3*A3^3*A1*A3*A1^2*A2I*A1I^2*A3I^3*A1I*A3I*A1I*A3I^3*A2^3*A1*A2I*A3^3*A1*A2^2*A1*A3*A1I*A3I^2*A1I*A3 - (1/z^4)*ID3;
A1I*A3I^3*A2^3*A3I*A1I*A2I^3*A3^3*A1*A3^3*A1^2*A2*A3*A2I^3*A3^3*A1*A3*A2I^3*A3^3*A1*A3*A1*A3^3*A1^2*A2 - (1/z^2)*ID3;

// In the abelianized group, these relations all reduce to multiples of 3*A1-2*A2+7*A3 = 0

// Calculations for Section 1.3.
// *****************************

// Calculations for Section 1.5.
// *****************************

F3:=FiniteField(3);
P3<x>:=PolynomialRing(F3);
F9<ii>:=ext< F3 | x^2+1 >;

MatsF9:=MatrixRing(F9,3);
GL3F9:=GeneralLinearGroup(3,F9);
ID3F9:=MatsF9!1;

F9conj:=hom< F9 -> F9 | -ii>;

F9starmat:=function(matvar);
return MatsF9![F9conj(matvar[1][1]),F9conj(matvar[2][1]),F9conj(matvar[3][1]),
               F9conj(matvar[1][2]),F9conj(matvar[2][2]),F9conj(matvar[3][2]),
               F9conj(matvar[1][3]),F9conj(matvar[2][3]),F9conj(matvar[3][3])];
end function;

// We define $\rho$ from the ring of matrices with entries in~$\Z[\zeta]$
// to the ring of matrices with entries in $F_9$ by mapping $\zeta$ to $ii$.

// Here are the images of $u'$, $v'$ and $b'$ under $\rho$ (in the notation of page 3):

rhoUD:=MatsF9![
ii^3+ii^2-ii,1-ii,0,
ii^3+ii^2-1,ii-ii^3,0,
0,0,1];

rhoVD:=MatsF9![
        ii^3,0,0,
ii^3+ii^2-ii-1,1,0,
          0,0,1];

rhoBD:=MatsF9![
               1,          0,       0,
-2*ii^3-ii^2+2*ii+2,ii^3+ii^2-ii-1,-ii^3-ii^2,
           ii^2+ii,     -ii^3-1,-ii^3+ii+1];

// Recall that $\gamma_0$ is the matrix used to conjugate matrices unitary
// with respect to Femptyset to matrices unitary with respect to Fdiag
// We conjugate by the image CC of $\gamma_0$ to get matrices
// with entries in $F_9$ which are unitary with respect to the usual form:
CC:=MatsF9![
1,0,0,
1,1,0,
0,0,1];

// That is, we form $\rho'$ by setting $\rho'(g)=CC*\rho(g)*CC^-1$.

rhodUD:=CC*rhoUD*CC^-1;
rhodVD:=CC*rhoVD*CC^-1;
rhodBD:=CC*rhoBD*CC^-1;

rhodJD:=(rhodUD*rhodVD)^2;

// The following are zero, checking that the images are unitary with respect to the usual form.
F9starmat(rhodUD)*rhodUD-ID3F9;
F9starmat(rhodVD)*rhodVD-ID3F9;
F9starmat(rhodBD)*rhodBD-ID3F9;

// So we get a homomorphism $\bar\Gamma\to PU(3,\F_9)$, also denoted $\rho'$. This 
// is surjective. In fact, let $\bar\Gamma^1$ denote the index 3 subgroup of $\bar\Gamma$
// consisting of the elements having a matrix representative with
// determinant 1. It is generated by $v$, $bu^{-1}$ and $u^{-1}b$.
// The image under $\rho'$ of $\bar\Gamma^1$ is all of $PU(3,\F_9)$,
// as we now check:
testgp:=sub< GL3F9 | rhodVD,rhodBD*rhodUD^-1,rhodUD^-1*rhodBD>;
#testgp; // output: 24192 =(6048 times 4)

// We now give an explicit subgroup $\langle R,M\rangle$ of order 21 in $PU(3,\F_9)$.
R:=MatsF9![
   ii, ii-1,  -ii,
    1,-ii-1,   -1,
-ii-1,    0,-ii-1];

M:=MatsF9![
 -ii,  ii,1-ii,
ii-1, -ii,  ii,
 -ii,ii-1, -ii];

// The following are zero:
R^7 - ID3F9;
M^3 - ID3F9;
M*R*M^2-R^2;

// Recall that the generators of $\Pi$ are called $a_1$, $a_2$ and $a_3$.
// The elements $a_i'=\gamma_0^{-1}a_i\gamma_0$ have entries in $\Z[\zeta]$.
// Here are their images under $\rho$:

rhoA1D:=MatsF9![
      ii^3+ii^2-1,  -ii^2+ii,       ii,
2*ii^3+2*ii^2-ii-2,-ii^3+ii+1, ii^2+ii+1,
        ii^3+ii^2,  -ii^2+1,-ii^3+ii+1];

rhoA2D:=MatsF9![
    ii^3+ii^2-1,-ii^3+ii-1,           1,
2*ii^3+2*ii^2-1,  -ii^2+1,-ii^3-ii^2+ii+1,
    ii^3+ii^2-1,       0,    -ii^3+ii+1];

rhoA3D:=MatsF9![
        ii^3-ii-1,      -ii^2+1, ii^2-ii-1,
2*ii^3+ii^2-2*ii-2,-ii^3-ii^2+ii+1, ii^3-ii-1,
       -ii^3+ii+1,       ii^2-1,-ii^3+ii+1];

// We then conjugate by CC
rhodA1D:=CC*rhoA1D*CC^-1;
rhodA2D:=CC*rhoA2D*CC^-1;
rhodA3D:=CC*rhoA3D*CC^-1;

// We now check that $\Pi={\rho'}^{-1}(\langle R,M\rangle)$. 
// The following are zero, showing that $\rho'(\Pi)$ is contained
// in $\langle R,M\rangle$.

M - (-rhodA1D*rhodA2D);
R - (-ii*rhodA2D);
R - (-rhodA3D^(-1));

// Now we see that $\Pi={\rho'}^{-1}(\langle R,M\rangle)$ since
// $\Pi$ had index 288 in $\bar\Gamma^1$ and $\langle R,M\rangle$ 
// has index 288 in $PU(3,\F_9)$.

// Calculations for Section 2.1.
// *****************************

// The method of finding the stated presentations of the groups $\bar\Gamma_0$
// and $\bar\Gamma_c$ is to describe the precise form of the matrices in these groups,
// and so to see that these groups are isomorphic to groups of 2 by 2 matrices which are
// unitary with respect to simple specific forms and have entries in $\Z[\zeta]$.
// We show that $\bar\Gamma_0$ is isomorphic to an index 3 subgroup of a group of 2 by 2 matrices called $U_0$,
// while $\bar\Gamma_c$ is isomorphic to an index 4 subgroup of a group of 2 by 2 matrices called $U_c$.
// Presentations of $U_0$ and $U_c$ may be found by various well-known methods. The quickest
// way is perhaps to use quaternion algebras, for which routines are available in Magma, 
// but there are more elementary methods.
// The details are given in a LaTeX file while is commented out below.
// Having found presentations of $U_0$ and $U_c$, presentations of their index~3 and index~4
// subgroups $\bar\Gamma_0$ and $\bar\Gamma_c$ may be found using Magma's Rewrite routine.
// Some fairly easy calculations show that the presentations thus found are equivalent to
// the presentations stated in Section 2.1

// Start of LateX file:
// ********************
// \documentclass[a4paper,reqno]{amsart}
// \usepackage[]{amsmath, amsthm, amsfonts, verbatim, amssymb}
// 
// \let\bbbibitem\bibitem
// \renewcommand{\bibitem}[2][]{\bbbibitem[#1]{#2}\label{#2}}
// 
// \hoffset=-6.5mm
// \textwidth=14cm
// \newcommand\C{{\mathbb C}}
// \newcommand\F{{\mathbb F}}
// \newcommand\Ham{{\mathbb H}}
// \newcommand\Q{{\mathbb Q}} 
// \newcommand\R{{\mathbb R}} 
// \newcommand\Z{{\mathbb Z}}
// \def\cO{{\mathcal O}}  
// \newcommand{\Adj}{\mathrm{Adj}}
// \newcommand\U{{\mathrm U}}
// \newcommand\SU{{\mathrm{SU}}}
// \newcommand\cZ{{\mathcal Z}}
// \begin{document}
// \newtheorem {lemm}{Lemma}
// \newtheorem {coro}{Corollary}
// %
// \title[Details about Section 2.1]
// {Details about Section 2.1}
// 
// \maketitle
// We next want to describe the groups $\bar\Gamma_c$ and~$\bar\Gamma_0$ of elements fixing~$M_c$ and~$M_0$,
// respectively.
// \begin{lemm}\label{lem:gfixesMalpha}For any $\alpha\in\C$, a $3\times3$ matrix $g=(g_{ij})$ with complex entries 
// which is unitary with respect to~$F$ satisfies $g(M_\alpha)=M_\alpha$ if and only if
// \begin{itemize}
// \item[(a)] $g_{13}=\alpha g_{23}$, and
// \item[(b)] $g_{12}=\alpha(\alpha g_{21}-g_{11}+g_{22})$.
// \end{itemize}
// \end{lemm}
// \begin{proof}This is straightforward.
// \end{proof}
// \begin{section}{The case $\bar\Gamma_0$.}
// 
// \begin{lemm}\label{lem:gamma0}If $gZ\in\bar\Gamma$, then $gZ\in\bar\Gamma_0$ if and
// only if we can write
// \begin{equation}\label{eq:gamma0matrix}
// g=\theta'
// \begin{pmatrix}1&0&0\\0&1&0\\0&0&\theta\end{pmatrix}
// \begin{pmatrix}
// 1&0&0\\
// 0&a&(r-1)b\\
// 0&\overline{b}&\overline{a}
// \end{pmatrix}
// \end{equation}
// where $a,b\in\Z[\zeta]$, $\theta,\theta'\in\{\zeta^k:k=0,\ldots,11\}$, $|a|^2-(r-1)|b|^2=1$,
// and $a-1\in(r-1)\Z[\zeta]$. This expression for~$g$ is unique, with $\theta'=g_{11}$ and ${\theta'}^3\theta=\det(g)$.
// \end{lemm}
// \begin{proof}Suppose that $gZ\in\bar\Gamma_0$. Applying Lemma~\ref{lem:gfixesMalpha} for $\alpha=0$ to $g$ and to
// \begin{equation}\label{eq:ginv}
// g^{-1}=F^{-1}g^*F=\begin{pmatrix}
// {\bar g_{11}}&{\bar g_{21}}&-(r-1){\bar g_{31}}\\
// {\bar g_{12}}&{\bar g_{22}}&-(r-1){\bar g_{32}}\\
// -{\bar g_{13}}/(r-1)&-{\bar g_{23}}/(r-1)&{\bar g_{33}}
// \end{pmatrix},
// \end{equation}
// we see that $g_{12}=g_{13}=g_{21}=g_{31}=0$. The condition that $\gamma_0^{-1}g\gamma_0$ has entries in~$\Z[\zeta]$
// tells us that $g_{11}$, $g_{22}$, $g_{33}$, $g_{32}$, $(g_{11}-g_{22})/(r-1)$ and 
// $g_{23}/(r-1)$ are in~$\Z[\zeta]$.
// 
// Now $g^*Fg=F$ implies that $|g_{11}|^2=1$. This and
// $g_{11}\in\Z[\zeta]$ implies that $g_{11}$ is a power of~$\zeta$, and
// so replacing $g$ by $g_{11}^{-1}g$, we may suppose that
// $g_{11}=1$. Also, $a=g_{22}$ and $b=g_{23}/(r-1)$ are
// in~$\Z[\zeta]$. Now
// $\det(g)=\det(\gamma_0^{-1}g\gamma_0)\in\Z[\zeta]$, and $g^*Fg=F$
// implies that $|\det(g)|=1$. So $\theta=\det(g)$ is also a power
// of~$\zeta$. Using the fact that $F^{-1}g^*F$ equals
// $\theta^{-1}\Adj(g)$, we see that $g_{33}=\bar a\det(g)$ and
// $g_{32}=\bar b\det(g)$, and then that $|a|^2-(r-1)|b|^2=1$. Finally,
// it is easy to check that $\gamma_0^{-1}g\gamma_0$ has entries
// in~$\Z[\zeta]$ if and only if $a-1\in(r-1)\Z[\zeta]$.
// \end{proof}
// Let $\U_0$ denote the group of matrices with entries in~$\Z[\zeta]$ which are 
// unitary with respect to
// \begin{displaymath}
// \begin{pmatrix}
// 1&0\\
// 0&1-r
// \end{pmatrix}.
// \end{displaymath}
// If $\SU_0$ is the subgroup of~$\U_0$ consisting of its elements
// of determinant~1, then $\U_0$ is the semidirect product of~$\SU_0$ and the group 
// of order~12 generated by the matrix $z=\begin{pmatrix}1&0\\0&\zeta\end{pmatrix}$.
// We define an embedding of~$\bar\Gamma_0$ into~$\U_0$ as follows. If $gZ\in\bar\Gamma_0$, 
// write~
// $g$ as in~\eqref{eq:gamma0matrix}, and set 
// \begin{equation}%\label{eq:phizero}
// \psi_0(gZ)=
// \begin{pmatrix}
// 1&0\\
// 0&\theta
// \end{pmatrix}
// \begin{pmatrix}
// a&(r-1)b\\
// \overline{b}&\overline{a}
// \end{pmatrix}.
// \end{equation}
// \begin{lemm}\label{lem:su0presentation}The group $\SU_0$ is generated by 
// \begin{displaymath}
// d=\begin{pmatrix}
// \zeta&0\\
// 0&\zeta^{-1}
// \end{pmatrix}
// \quad\text{and}\quad
// x=\begin{pmatrix}
// \zeta^3+\zeta^2-1&-\zeta^2-\zeta+1\\
// \zeta^3+\zeta^2-1&-\zeta^3-\zeta^2
// \end{pmatrix},
// \end{displaymath}
// and has the following presentation with respect to these generators:
// \begin{displaymath}
// \langle d,x\ |\ d^{12}=x^3=1, (dx^2)^3=d^6, d^6x=xd^6\rangle.
// \end{displaymath}
// We get a presentation of~$\U_0$ by adding the generator $z$ and
// the relations $z^{12}=1$, $zdz^{-1}=d$ and $zxz^{-1}=d^6x^{-1}d$. The subgroup 
// $H_0=\langle xd,dx,d^3\rangle$ of~$\SU_0$ has index~3.
// \end{lemm}
// \begin{proof}Given a field~$\F$ not of characteristic~2, and $\alpha,\beta\in\F^\times$,
// the {\it quaternion algebra\/} $(\alpha,\beta)_\F$ consists of elements
// $\xi=x_0+x_1i+x_2j+x_3k$, where $x_0,\ldots,x_3\in\F$, with an associative multiplication 
// satisfying $ij=k=-ji$ and $i^2=\alpha$, $j^2=\beta$. The {\it reduced norm\/} $N(\xi)=N_A(\xi)$ of~$\xi$ is
// $x_0^2-\alpha x_1^2-\beta x_2^2+\alpha\beta x_3^2$. If $\xi,\xi'\in A$, then $N(\xi\xi')=N(\xi)N(\xi')$.
// Writing $a=x_0+x_1i$ and $b=x_2+x_3i$, we can think of $(\alpha,\beta)_\F$ as consisting of elements $a+bj$, 
// where $a,b\in\F(\sqrt{\alpha})$, $j^2=\beta$, and $ja=\overline{a}j$ for the automorphism 
// $\overline{\phantom{a}}:x_0+x_1i\mapsto x_0-x_1i$ of~$\F(\sqrt{\alpha})$. The classical Hamiltonian quaternion algebra is $\Ham=(-1,-1)_\R$. 
// We have $N_\Ham(a+bj)=|a|^2+|b|^2$ for $a,b\in\R(\sqrt{-1}\,)=\C$.
// 
// Let $A=(-1,r-1)_{\Q(r)}$. Identifying $i\in A$ with $\zeta^3\in\ell=\Q(\zeta)$, we see that
// $A=\{a+bj:a,b\in\ell\}$, and that $N(a+bj)=|a|^2-(r-1)|b|^2$. Let $\cO=\{a+bj\in A:a,b\in\Z[\zeta]\}$.
// Then $\cO$ is a subring of~$A$, closed under (left) multiplication by~$\Z[r]$, and so is an {\it order\/}
// in~$A$. In fact, it is a {\it maximal\/} order.
// Clearly $\SU_0$ is isomorphic to the group $\cO^1$ of elements of~$\cO$ having reduced norm~1.
// The group $\Z[r]^\times$ of units in~$\Z[r]$ consists of the elements $m+nr$, where $m,n\in\Z$ 
// and $m^2-3n^2=\pm1$. Now $m^2-3n^2=-1$ never holds, and $m^2-3n^2=1$ if and only if $m+nr=(2+r)^k$ 
// for some $k\in\Z$~(see \cite[\S7.8]{NZM}, for example). So $\Z[r]^\times$ is generated by~$-1$ and $2+r$. 
// If $\xi\in\cO^\times$, then $N(\xi)\in\Z[r]^\times$. 
// In fact, $N(\xi)$ is never equal to~$-1$, 
// for if $\epsilon:\Q(r)\to\R$ is the field embedding mapping $r$ to~$-\sqrt{3}$, then
// \begin{displaymath}
// f:x_0+x_1i+x_2j+x_3k\mapsto\epsilon(x_0)+\epsilon(x_1)\,i+\epsilon(x_2)\sqrt{\sqrt{3}+1}\;j+\epsilon(x_3)\sqrt{\sqrt{3}+1}\;k
// \end{displaymath}
// is an embedding of~$A$ into~$\Ham$ satisfying $\epsilon(N_A(\xi))=N_\Ham(f(\xi))$.
// % To see a check of this, see the reduce syntax file hamiltonian_embeddings.red.
// Now $\cO^1\subset\cO^\times$. Since $2+r=N(\zeta+1)$, if $\xi\in\cO^\times$ and $N(\xi)=(2+r)^k$, 
// then $\xi/(\zeta+1)^k\in\cO^1$. Since $(\zeta+1)^2=\zeta(2+r)$, we see that $\cO^1/\{1,-1\}$ 
// embeds as an index~2 subgroup of~$\cO^\times/\Z[r]^\times$. Magma has routines for finding a 
// presentation of~$\cO^\times/\Z[r]^\times$. As these may be less familiar to the reader, we give some
// details. We set up $\Q(\zeta)$, $\Q(r)$ and~$A$ with the commands
// \small{
// \begin{displaymath}
// \begin{aligned}
// &\mathtt{L\langle z\rangle:=CyclotomicField(12);}\phantom{XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX}\\
// &\mathtt{K\langle r\rangle:=sub\langle\; L\ |\ z+1/z\rangle;}\\
// &\mathtt{A\langle i,j,k\rangle:=QuaternionAlgebra\langle\; K\ |\ -1,\;r-1\rangle;}
// \end{aligned}
// \end{displaymath}
// }
// As $\zeta=(r+\zeta^3)/2$, we set $\mathtt{zz:=(r+i)/2;}$ and $\mathtt{O:=Order([1,zz,j,zz{*}j]);}$. Now
// the commands $\mathtt{G:=FuchsianGroup(O);}$ and $\mathtt{u,m:=Group(G);}$ and $\mathtt{u;}$ give
// a presentation for $\cO^\times/\Z[r]^\times$. The command $\mathtt{[A!Quaternion(m(U.i)):i\ in\ [1..2]];}$
// makes the generators $u_1$, $u_2$ explicit. We find that $u_1=(2+r-i)/2$ and $u_2=-(r+2)(i+k)$. These satisfy
// $u_1^{12}=u_2^2=(u_1u_2)^3=1$ (mod~$\Z[r]^\times$). Note that $N(u_1)=N(u_2)=2+r$. Magma verifies that
// the subgroup of the abstract group $\langle u_1,u_2\ |\ u_1^{12}=u_2^2=(u_1u_2)^3=1\rangle$ has a single
// index~2 subgroup, and it is generated by $g_1=u_2u_1^{-1}$ and~$g_2=u_1^2$, and the relations
// $g_1^3=(g_1g_2)^3=g_2^6=1$ give a presentation. For the given concrete $u_1,u_2\in A$, we
// set $g_1=u_2u_1^{-1}$ and $g_2=(2-r)u_1^2$. Then $g_1,g_2\in\cO^1$ generate $\cO^1/\{-1,1\}$
// and satisfy $g_1^3=1$, $(g_1g_2)^3=-1=g_2^6$. The given elements~$d$ and~$x$ are just $g_2^{-1}$
// and $g_2^{-3}g_1g_2^{-3}$. So they and the given relations form a presentation of~$\SU_0$.
// 
// The remaining assertions are routine to verify.
// \end{proof}
// \begin{lemm}\label{lem:imageofpsi0}The image under~$\psi_0$ of~$\bar\Gamma_0$
// is~$\langle z\rangle H_0$.
// \end{lemm}
// \begin{proof}The elements $xd$, $dx$, $d^3$ and~$z$ are all in $\psi_0(\bar\Gamma_0)$, being respectively
// the images of the elements $gZ$ of~$\bar\Gamma_0$ for the following~$g$'s:
// \begin{displaymath}
// \zeta^{-4}j^{-3}bj^7,\quad\zeta^{-4}j^{-1}bj^5,\quad\zeta^{-3}v^{-1}j^6,\quad\text{and}\ \zeta j^{-1}.
// \end{displaymath}
// Now $d\not\in\psi_0(\bar\Gamma_0)$ since $\zeta-1\not\in(r-1)\Z[\zeta]$. So 
// we have $\langle z\rangle H_0\subset\psi_0(\bar\Gamma_0)\subsetneqq \U_0$. Since $H_0$ has index~3 in~$\SU_0$
// we have $\langle z\rangle H_0=\psi_0(\bar\Gamma_0)$. 
// \end{proof}
// \begin{coro}The elements $s_2=(jb)^{-1}$, $s_3=b$, $s_{12}=j$ and $z_0=v$ generate 
// $\bar\Gamma_0$, and together with the relations
// \begin{displaymath}
// s_{12}^{12}=s_3^3=1,s_2^2=z_0^3,z_0^4=[s_{12},z_0]=[s_3,z_0]=[s_2,z_0]=s_{12}s_3s_2=1.
// \end{displaymath}
// form a presentation of~$\bar\Gamma_0$.
// \end{coro}
// \begin{proof}Let $\bar\Gamma_{\mathrm{ce}}$ denote the abstract finitely presented group
// on generators $s_2$, $s_3$, $s_{12}$ and~$z_0$ and the stated relations. There is a homomorphism
// $\bar\Gamma_{\mathrm{ce}}\to\bar\Gamma_0$ mapping $s_2$ to~$(jb)^{-1}$, $s_3$ to~$b$, $s_{12}$ to~$j$
// and $z_0$ to~$v$. The last lemma allows us to identify $\U_0$ with the abstract finitely presented
// group with generators $d$, $x$ and~$z$ and the above-stated relations. Magma's {\tt Rewrite(G,H)\/} command, applied to this finitely presented
// group and its subgroup $H=\psi_0(\bar\Gamma_0)$ (which has index~3 in~$\U_0$), gives a presentation of~$H$,
// with generators $R_1=dx$ and $R_2=xzx^{-1}$. Now $R_1=\psi_0(s_{12}^{-1}s_3s_{12}^5)$
// and $R_2=\psi_0(s_{12}^{-2}s_3s_{12}^5z_0^2)$. Thus $\bar\Gamma_0$ is generated by $s_2$, $s_3$, $s_{12}$ and~$z_0$.
// The {\tt Rewrite\/} command gives relations involving words in $R_1$ and~$R_2$. Using
// $R_1=\psi_0(s_{12}^{-1}s_3s_{12}^5)$ and $R_2=\psi_0(s_{12}^{-2}s_3s_{12}^5z_0^2)$, we can express 
// these words as words in $s_2$, $s_3$, $s_{12}$ and~$z_0$, and it is routine to verify that
// these words equal~1 is a consequence of the stated relations of the central extension. This
// means that there is a homomorphism $H\to\bar\Gamma_{\mathrm{ce}}$ mapping $R_1$ and~$R_2$
// to~$s_{12}^{-1}s_3s_{12}^5$ and~$s_{12}^{-2}s_3s_{12}^5z_0^2$ respectively, and that the homomorphisms
// $\bar\Gamma_{\mathrm{ce}}\to\bar\Gamma_0\to H$ are isomorphisms.\end{proof}
// 
// \end{section}
// 
// \begin{section}{The case $\bar\Gamma_c$.}
// We now describe $\bar\Gamma_c$. Recall that $c=(r-1)(\zeta^3-1)/2=\zeta^2-\zeta$.
// 
// \begin{lemm}\label{lem:gammac}If $gZ\in\bar\Gamma$, then $gZ\in\bar\Gamma_c$ if and
// only if we can write
// \begin{equation}\label{eq:gammacmatrix}
// g=\theta'
// \begin{pmatrix}1&0&0\\0&1&0\\0&0&\theta\end{pmatrix}
// \begin{pmatrix}
// (a(2-r)+1)/(3-r)&c(a-1)/(3-r)&bc\\
// (a-1)\overline{c}/(3-r)&(a+2-r)/(3-r)&b\\
// \overline{b}\overline{c}/(r-1)&\overline{b}/(r-1)&\overline{a}
// \end{pmatrix}
// \end{equation}
// where $a,b\in\Z[\zeta]$, $\theta,\theta'\in\{\zeta^k:k=0,\ldots,11\}$, $|a|^2-r|b|^2=1$,
// and $a-1\in r\Z[\zeta]$. This expression for~$g$ is unique, with $\theta'=g_{11}-cg_{21}$ and ${\theta'}^3\theta=\det(g)$.
// \end{lemm}
// \begin{proof}Suppose that $gZ\in\bar\Gamma_0$. Applying Lemma~\ref{lem:gfixesMalpha} for $\alpha=c$ to~$g$ and to~$g^{-1}$, we have
// $g_{13}=cg_{23}$, $g_{12}=c(cg_{21}-g_{11}+g_{22})$,
// $\overline{g_{31}}=c\overline{g_{32}}$, and $\overline{g_{21}}=c(c\overline{g_{12}}-\overline{g_{11}}+\overline{g_{22}})$.
// From the second and fourth of these equations, we find that $\overline{c}g_{12}=cg_{21}$.
// 
// Using Lemma~\ref{lem:gfixesMalpha} again, we see that the map $g\mapsto g_{11}-cg_{21}$ 
// is multiplicative on the group of matrices satisfying $g(M_c)=M_c$.
// So we get
// $1=\bigl(g_{11}-cg_{21}\bigr)\bigl(\overline{g_{11}}-c\overline{g_{12}}\bigr)
// =|g_{11}-cg_{21}|^2$ by applying this to $g$ and~$g^{-1}$, and so $\theta'=g_{11}-cg_{21}$ has modulus~1. The condition that $\gamma_0^{-1}g\gamma_0$ 
// has entries in~$\Z[\zeta]$ implies in particular that $g_{11},g_{21}\in\frac{1}{r-1}\Z[\zeta]$,
// so that $\theta'\in\frac{1}{r-1}\Z[\zeta]$. This and $|\theta'|=1$ imply that 
// $\theta'\in\{\zeta^k:k=0,\ldots,11\}$. So replacing $g$ by ${\theta'}^{-1}g$, if necessary, we
// may suppose that $g_{11}-cg_{21}=1$. We can now express $g_{11}$, $g_{12}$ and $g_{21}$ in terms 
// of~$g_{22}$. Now let $N=F^{-1}g^*F-{\Adj}(g)/\theta=(n_{ij})$, where $\theta=\det(g)$. By~\eqref{eq:ginv},
// this is zero. We solve $cn_{11}+n_{12}=0$ for $g_{22}$, obtaining $g_{22}=(|c|^2+\theta\overline{g_{33}})/(|c|^2+1)$.
// Now solving $n_{31}=0$, we get $g_{32}=\overline{g_{23}}/((r-1){\bar\theta})=\theta\overline{g_{23}}/(r-1)$, 
// using $|\theta|=1$. Write $a=\theta\overline{g_{33}}$ and $b=g_{23}$. Then \eqref{eq:gammacmatrix} holds.
// There is just one remaining condition on $a$ and~$b$ to ensure that
// $N=0$, namely $|a|^2-r|b|^2=1$. This equation is also the condition that the determinant of the last
// matrix on the right in~\eqref{eq:gammacmatrix} is~1. So taking determinants, we see that 
// $\det(g)={\theta'}^3\theta$. As in Lemma~\ref{lem:gamma0}, $\det(g)\in\Z[\zeta]$, and so 
// $\theta\in\{\zeta^k:k=0,\ldots,11\}$ too. Finally, by considering $g-I$, it is routine
// to check that $\gamma_0^{-1}g\gamma_0$ has entries in~$\Z[\zeta]$ if and only if $a-1\in r\Z[\zeta]$.
// \end{proof}
// 
// Let $\U_c$ be the group of matrices with entries in~$\Z[\zeta]$ which are unitary
// with respect to
// \begin{displaymath}
// \begin{pmatrix}
// 1&0\\
// 0&-r
// \end{pmatrix}.
// \end{displaymath}
// If $\SU_c$ is the subgroup of~$\U_c$ consisting of its elements
// of determinant~1, then $\U_c$ is the semidirect product of~$\SU_c$ and the group 
// of order~12 generated by the above matrix~$z$.
// We define an embedding of~$\bar\Gamma_c$ into~$\U_c$ as follows. If $gZ\in\bar\Gamma_c$, write
// $g$ as in~\eqref{eq:gammacmatrix}, and set 
// \begin{equation}%\label{eq:phizero}
// \psi_c(gZ)=
// \begin{pmatrix}
// 1&0\\
// 0&\theta
// \end{pmatrix}
// \begin{pmatrix}
// a&rb\\
// \overline{b}&\overline{a}
// \end{pmatrix}.
// \end{equation}
// Then $\psi_c$ is an injective homomorphism  $\bar\Gamma_c\to \U_c$.
// 
// \begin{lemm}\label{lem:piclemma}The group $\SU_c$ has generators
// \begin{displaymath}
// d=\begin{pmatrix}\zeta&0\\0&\zeta^{-1}\end{pmatrix},\ 
// q=\begin{pmatrix}r+1&r/c\\1/\overline{c}&r+1\end{pmatrix},\ \text{and}\ 
// s=\begin{pmatrix}\zeta^3(r+1)&r/c\\1/\overline{c}&\zeta^{-3}(r+1)\end{pmatrix},
// \end{displaymath}
// and has the following presentation with respect to these generators:
// \begin{displaymath}
// \SU_c=\langle d,q,s\ |\ d^{12}=1,s^2=(qd^3)^2=(qd^2sd^2)^2=d^6\rangle.
// \end{displaymath}
// A presentation for $\U_c$ is obtained to adding to the above presentation of~$\SU_c$ the generator 
// $z$ and the relations $z^{12}=1$, $zd=dz$, $zsz^{-1}=dqd^2$ and $zqz^{-1}=d^{-2}sd^{-1}$.
// The subgroup $H_c=\langle sd,ds,q\rangle$ has index~4 in~$\SU_c$.
// \end{lemm}
// \begin{proof}The proof is similar to that of Lemma~\ref{lem:su0presentation}. We
// use the quaternion algebra $A=(-1,r)_{\Q(r)}$ and the maximal order $\cO=\{a+bj:a,b\in\Z[\zeta]\}$.
// Since $N(a+bj)=|a|^2-r|b|^2$, we have $\SU_c\cong\cO^1$. Again $\cO^1/\{-1,1\}$ embeds as a
// subgroup of index~2 in~$\cO^\times/\Z[r]^\times$ (we exclude $N(\xi)=-1$ in the same way, with
// $3^{1/4}$ in place of $\sqrt{\sqrt{3}+1}$ in the definition of the embedding $A\to\Ham$). 
// This time we get a presentation
// for $\cO^\times/\Z[r]^\times$ with generators $u_1=(r+2-i)/2$ and~$u_2=(r+1-(3r+5)i-2(r+2)k)/2$
// satisfying $u_1^{12}=u_2^4=(u_1u_2)^2=1$ (mod~$\Z[r]^\times$). The elements $g_1=u_2u_1^{-1}$,
// $g_2=u_1^2$ and $g_3=u_1u_2$ generate one of the three index~2 subgroups of the abstract group
// $\langle u_1,u_2\ |\ u_1^{12}=u_2^4=(u_1u_2)^2=1\rangle$, and this subgroup
// has presentation $(g_1g_2)^2=(g_1g_3)^2=g_3^2=g_2^6=1$. For the given concrete $u_1,u_2\in A$, we
// set $g_1=u_2u_1^{-1}$ and $g_2=(2-r)u_1^2$ and $g_3=(2-r)u_1u_2$. Then $g_1,g_2,g_3\in\cO^1$ generate 
// $\cO^1/\{-1,1\}$ and satisfy $(g_1g_2)^2=(g_1g_3)^2=g_3^2=g_2^6=-1$. We have $g_1=-ds$, $g_2=d^{-1}$
// and~$g_3=-d^2qd$. The result follows.
// \end{proof}
// 
// \begin{lemm}\label{lem:imageofpsic}The image of $\bar\Gamma_c$ in~$\U_c$ is~$\langle z\rangle H_c$.
// \end{lemm}
// \begin{proof}Now $H_c\subset\psi_c(\bar\Gamma_c)$, since for the following elements $g$ of~$\bar\Gamma$:
// \begin{displaymath}
// \zeta^{-4}j^7bu^{-1}bu j^7,\quad\zeta^{-4}j^{-3}bu^{-1}bu j^5,\quad j^4bu^{-1}bu^{-1}j^2,
// \end{displaymath}
// we have $\det(g)=1$, $g_{11}-cg_{21}=1$, $g_{13}=cg_{23}$ and $g_{12}=c(cg_{21}-g_{11}+g_{22})$, while $\psi_c(gZ)$ 
// equals $sd$, $ds$ and~$q$, respectively. Also, $z=\psi_c(gZ)$ for $g=\zeta j^{-1}$. Hence $\langle z\rangle H_c\subset\psi_c(\bar\Gamma_c)$.
// Now $d,d^2,d^3\not\in\psi_c(\bar\Gamma_c)$, since $\zeta^i-1\not\in r\Z[\zeta]$ for $i=1,2,3$,
// and so the index of~$\psi_c(\bar\Gamma_c)$ in~$\U_c$ is at least~4. Since $[\SU_c:H_c]=4$, we must have
// $\psi_c(\bar\Gamma_c)=\langle z\rangle H_c$.
// \end{proof}
// \begin{coro}The elements $t_2=(bu^{-1})^2$, $t_4=j^{-1}(bu^{-1})^2$, $t_{12}=j$ and $z_c=u$
// $\bar\Gamma_c$, and together with the relations
// \begin{displaymath}
// t_{12}^{12}=1,t_4^4=z_c,t_2^2=z_c^3=[t_{12},z_c]=[t_4,z_c]=[t_2,z_c]=t_{12}t_4t_2=1.
// \end{displaymath}
// form a presentation of~$\bar\Gamma_c$.
// \end{coro}
// \begin{proof}Let $\bar\Gamma_{\mathrm{ce}}$ denote the abstract finitely presented group
// on generators $t_2$, $t_4$, $t_{12}$ and~$z_c$ and the stated relations. There is a homomorphism
// $\bar\Gamma_{\mathrm{ce}}\to\bar\Gamma_c$ mapping $t_2$ to~$(bu^{-1})^2$, $t_4$ to~$j^{-1}(bu^{-1})^2$, $t_{12}$ to~$j$
// and $z_c$ to~$u$. The last lemma allows us to identify $\U_c$ with the abstract finitely presented
// group with generators $d$, $q$, $s$ and~$z$ and the above-stated relations. Magma's {\tt Rewrite(G,H)\/} command, applied to this finitely presented
// group and its subgroup $H=\psi_c(\bar\Gamma_c)$ (which has index~4 in~$\U_c$), gives a presentation of~$H$,
// with generators $R_1=z$, $R_2=dqd^{-1}$ and $R_3=s^2d^{-2}$. Now $R_1=\psi_c(t_{12}^{-1})$, $R_2=\psi_c(t_{12}^6t_2)$
// and $R_3=\psi_c(z_c^{-1}t_{12}^8)$. Thus $\bar\Gamma_c$ is generated by $t_2$, $t_4$, $t_{12}$ and~$z_c$.
// The {\tt Rewrite\/} command gives relations involving words in $R_1$, $R_2$ and~$R_3$. Using
// $R_1=\psi_c(t_{12}^{-1})$, $R_2=\psi_c(t_{12}^6t_2)$ and $R_3=\psi_c(z_c^{-1}t_{12}^8)$, we can express 
// these words as words in $t_2$, $t_4$, $t_{12}$ and~$z_c$, and it is routine to verify that
// these words equal~1 is a consequence of the stated relations of the central extension. This
// means that there is a homomorphism $H\to\bar\Gamma_{\mathrm{ce}}$ mapping $R_1$, $R_2$ and~$R_3$
// to~$t_{12}^{-1}$, $t_{12}^6t_2$ and~$z_c^{-1}t_{12}^8$ respectively, and that the homomorphisms
// $\bar\Gamma_{\mathrm{ce}}\to\bar\Gamma_c\to H$ are isomorphisms.\end{proof}
// \end{section}
// \begin{thebibliography}{12mm}
// \bibitem[NZM]{NZM}Niven, I., Zuckermann H. S., Montgomery, H. L,  An Introduction to the Theory of Numbers,
// (Fifth Edition), Wiley, 1991.
// \end{thebibliography}
// \end{document}
// end of Latex File.
// ******************

// The presentation of the group $\bar\Gamma_0$:
// *********************************************

// The following elements s2, s3, s12 and z0 all satisfy these conditions, 
// and so are in $\bar\Gamma_0$.

s2:=(J*B)^-1;
s3:=B;
s12:=J;
z0:=V;

// The following are zero, checking that s2,s3,s12 and z0 satisfy
// the stated relations of $\bar\Gamma_0$:
s2^2-z*z0^3;
s3^3-ID3;
s12^12-ID3;
z0^4-ID3;
s12*s3*s2-ID3;
s2*z0-z0*s2;
s3*z0-z0*s3;
s12*z0-z0*s12;
// This means that there is a homomorphism from the abstract group with
// the presentation stated for $\bar\Gamma_0$ in Section 2.1 to the group $\bar\Gamma_0$.
// See the commented out LateX file for the definition of the group $U_0$ and
// the injective homomorphism $\psi_0:\bar\Gamma_0\to U_0$. As proved in that file,
// the image of $\bar\Gamma_0$ under~$\psi_0$ has index 3 in $U_0$.
// Here we set up that map $\psi_0$:
Mats2L:=MatrixRing(L,2);

Fzero:=Mats2L![1,0,
               0,1-r];

FzeroI:=Mats2L![1,0,
                0,-(r+1)/2];

ID2:=Mats2L!1;

starmat2:=function(matvar);
return Mats2L![hLL(matvar[1][1]),hLL(matvar[2][1]),
               hLL(matvar[1][2]),hLL(matvar[2][2])];
end function;


// Here are the generators of U_0:
X:=Mats2L![z^3+z^2-1,-z^2-z+1,
           z^3+z^2-1,-z^3-z^2];

D:=Mats2L![z,0,
           0,z^-1];

ZMAT:=Mats2L![1,0,
              0,z];
// Here are their inverses:
XI:=X^-1;
DI:=D^-1;
ZMATI:=ZMAT^-1;

// Checking that d, z and ZMAT are unitary with respect to Fzero:

// The following are zero:
starmat2(D)*Fzero*D-Fzero;
starmat2(X)*Fzero*X-Fzero;
starmat2(ZMAT)*Fzero*ZMAT-Fzero;

// Here is the function $\psi_0$ described above
// It maps elements of $\bar\Gamma_0$ to U_0:

psizero:=function(matvar);
 if ((matvar[1][1] eq 0) or (matvar[1][2] ne 0) or (matvar[1][3] ne 0) or (matvar[2][1] ne 0) or (matvar[3][1] ne 0)) 
  then print "Error - matrix does not fix M_0"; return matvar; 
 end if;
tmpmat:=(1/matvar[1][1])*matvar;
return Mats2L![tmpmat[2][2],tmpmat[2][3],tmpmat[3][2],tmpmat[3][3]];
end function;

// The following are zero:
psizero(z^-4*J^-3*B*J^7) - X*D;
psizero(z^-4*J^-1*B*J^5) - D*X;
psizero(z^-3*V^-1*J^6) - D^3;
psizero(z*J^-1)- ZMAT;

// See the LateX file for how we find a presentation for $U_0$:

// The group $U_0$ is generated by the above matrices X, D and ZMAT, and
// has the presentation
// $\langle d,x,z | d^{12}=x^3=z^12=1, (dx^2)^3=d^6, d^6x=xd^6, zdz^-1=d, zxz^-1= d^6x^{-1}d\rangle$.

// We set up an abstract finitely presented group with this presentation:
// FU0<xd,xx,xz>:=FreeGroup(3);
// relnlistU0:={
// xd^12,
// xx^3,
// (xd*xx^2)^3*xd^-6,
// xd^6*xx*xd^-6*xx^-1,
// xz^12,
// xz*xd*xz^-1*xd^-1,
// xz*xx*xz^-1*xd^-1*xx*xd^-6};
// 
// U0gp,phi0:=quo< FU0 | relnlistU0>;
// 
// d:=phi0(xd);
// x:=phi0(xx);
// zabstract:=phi0(xz);
// 
// // We know that $\psi_0:\bar\Gamma_0\to U_0$ is an injective homomorphism. We 
// // identify $U_0$ with the above finitely presented group U0gp.
// // Now $\psi_0$ has image containing $xd$, $dx$, $d^3$ and $zabstract$.
// // We check that the subgroup generated by these four elements is normal and of index 3:
// testsubgroup:=sub< U0gp | x*d,d*x,d^3,zabstract>;
// print "the index of testsubgroup in U0gp is", Index(U0gp,testsubgroup); // output: 3
// print "it is",IsNormal(U0gp,testsubgroup),"that testsubgroup is normal in U0gp";
// // We know that $\psi_0(\bar\Gamma_0)$ is a proper subgroup of $U_0$ because, for
// // example, the matrix $X$ has $(1,1)$ entry $a=\zeta^3+\zeta^2-1$, which does not
// // satisfy the condition $a-1\in(r-1)\Z[\zeta]$ which must be satisfied by the $(1,1)$
// // entry of any matrix in the image of $\bar\Gamma_0$ of $\bar\Gamma_0$. Since
// // $\psi_0(\bar\Gamma_0)$ contains the index 3 testsubgroup, we must have $\psi_0(bar\Gamma_0)=testsubgroup$.
// 
// psi0bargamma0:=testsubgroup;
// 
// // To get a presentation for the index 3 subgroup, and hence of $\bar\Gamma_0$, we using following Magma command:
// R:=Rewrite(U0gp,psi0bargamma0);
// print R;
// 
// // // Output:
// // Finitely presented group R on 2 generators
// // Generators as words in group U0gp
// // R1=U0gp.1*U0gp.2
// // R2=U0gp.2*U0gp.3*U0gp.2^-1
// // Relations
// // R2*R1*R2^-2*R1*R2^-1*R1*R2*R1^-1*R2=Id(R)
// // R2*R1^-1*R2*R1*R2^-1*R1*R2^-2*R1*R2=Id(R)
// // R2^-1*R1*R2*R1^-2*R2^2*R1^-1*R2^-1*R1^2*R2^-1=Id(R)
// // R2^12=Id(R)
// // R2^2*R1*R2^-3*R1*R2^2*R1*R2^-1*R1^-1*R2^-1*R1*R2=Id(R)
// //
// // U0gp.1, U0gp.2 and U0gp.3 here are d, x and zabstract, respectively.
// // So the index 3 subgroup is generated by the following
// // two elements, which are images under $\psi_0$ of the
// // explicit elements of $\bar\Gamma_0$:
// R1:=D*X;
// R2:=X*ZMAT*X^-1;
// 
// // The following are zero:
// psizero(z^-4*J^-1*B*J^5) - D*X;
// psizero(z^3*J^10*B*V^2*J^5) - X*ZMAT*X^-1;
// 
// // In terms of the given generators $s_2=(jb)^{-1}$, $s_3=b$, $s_{12}=j$ and $z_0=v$ of the central extension,
// // we have
// psizero(z^-4*s12^-1*s3*s12^5) - D*X;
// psizero(z^3*s12^10*s3*z0^2*s12^5) - X*ZMAT*X^-1;
// 
// // We now replace R1 and R2 in the five relations found above by the Rewrite routine
// // by s12^-1*s3*s12^5 and s12^10*s3*z0^2*z12^5, respectively, and check that they
// // are consequences of the stated relations amongst s_2, s_3, s_{12} and z_0 of the central extension.
// // This means that there is a homomorphism from $\psi_0(\bar\Gamma_0)$ to the central extension
// // mapping R1 to s12^-1*s3*s12^5 and R2 to s12^10*s3*z0^2*s12^5
// // write
// s3i:=s3^-1;
// s12i:=s12^-1;
// 
// s12^10*s3*z0^2*s12^5*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*z0^2*s12^5*s12^7*s3i*s12*s12^10*s3*z0^2*s12^5;
// s12^10*s3*z0^2*s12^5*s12^7*s3i*s12*s12^10*s3*z0^2*s12^5*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*z0^2*s12^5;
// s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*z0^2*s12^5*s12^7*s3i*s12*s12^7*s3i*s12*s12^10*s3*z0^2*s12^5*s12^10*s3*z0^2*s12^5*s12^7*s3i*s12*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2;
// (s12^10*s3*z0^2*s12^5)^12;
// s12^10*s3*z0^2*s12^5*s12^10*s3*z0^2*s12^5*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2*s12^7*z0^2*s3i*s12^2*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*z0^2*s12^5*s12^10*s3*z0^2*s12^5*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2*s12^7*s3i*s12*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*z0^2*s12^5;
// // Showing that these five relations are consequences of the relations given for the central extension:
// // First relation:
// s12^10*s3*z0^2*s12^5*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*z0^2*s12^5*s12^7*s3i*s12*s12^10*s3*z0^2*s12^5;
// //-> using z0 cental and z0^4=1
// s12^10*s3*s12^5*s12i*s3*s12^5*s12^7*s3i*s12^2*s12^7*s3i*s12^2*s12i*s3*s12^5*s12^7*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*s12^5*s12^7*s3i*s12*s12^10*s3*s12^5;
// //-> using s12^12=1 and s12i=s12^-1:
// s12^10*s3*s12^4*s3*s3i*s12^2*s12^7*s3i*s12*s3*s3i*s12*s3*s12^3*s3*s3i*s12^11*s3*s12^5;
// //-> using s3*s3i=1
// s12^10*s3*s12^4*s12^2*s12^7*s3i*s12*s12*s3*s12^3*s12^11*s3*s12^5;
// //-> using s12^12=1
// s12^10*s3*s12*s3i*s12^2*s3*s12^2*s3*s12^5;
// //-> conjugating by s12^5
// s12^3*s3*s12*s3i*s12^2*s3*s12^2*s3;
// //-> using s3i=s2*s12
// s12^3*s3*s12*s2*s12*s12^2*s3*s12^2*s3;
// //-> using s12*s3=s2^-1=s2*z0
// s12^2*s2*z0*s12*s2*s12^2*s2*z0*s12*s2*z0;
// //-> using z0 central
// s12^2*s2*s12*s2*s12^2*s2*s12*s2*z0^3;
// //-> using s2*s12=s3i
// s12^2*s3i*s3i*s12*s3i*s2*z0^3;
// //-> using s3^3=1
// s12^2*s3*s12*s3*s3*s2*z0^3;
// //-> conjugating by s12^-1 and using z0 central:
// s12*s3*s12*s3*s3*s2*s12*z0^3;
// //-> using s3*s2*s12=1
// s12*s3*s12*s3*z0^3;
// //-> using s12*s3=s2^-1:
// s2^-1*s2^-1*z0^3;
// // using s2^2=z0^3
// ID3;
// 
// // Second relation:
// s12^10*s3*z0^2*s12^5*s12^7*s3i*s12*s12^10*s3*z0^2*s12^5*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*z0^2*s12^5;
// //-> using z0 central and z0^4=1
// s12^10*s3*s12^5*s12^7*s3i*s12*s12^10*s3*s12^5*s12i*s3*s12^5*s12^7*s3i*s12^2*s12i*s3*s12^5*s12^7*s3i*s12^2*s12^7*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*s12^5;
// //-> using s12^12=1
// s12^10*s3*s3i*s12^11*s3*s12^4*s3*s3i*s12*s3*s3i*s12^9*s3i*s12*s3*s12^3*s3*s12^5;
// //-> using s3*s3i=1
// s12^10*s12^11*s3*s12^4*s12*s12^9*s3i*s12*s3*s12^3*s3*s12^5;
// //-> using s12^12=1
// s12^9*s3*s12^2*s3i*s12*s3*s12^3*s3*s12^5;
// //-> conjugating by s12^5
// s12^2*s3*s12^2*s3i*s12*s3*s12^3*s3;
// //-> using s3i=s3^2
// s12* s12*s3 *s12* s12*s3 *s3* s12*s3 *s12^2* s12*s3;
// // -> using s12*s3=s2^-1=s2*z0
// s12*s2*z0*s12*s2*z0*s3*s2*z0*s12^2*s2*z0;
// //-> using z0 central and z0^4=1
// s12*s2*s12*s2*s3*s2*s12^2*s2;
// //-> using s3*s2*s12=1
// s12*s2*s12*s2*s12*s2;
// //-> using s12*s2=s3i
// s3i*s3i*s3i;
// //-> using s3^3=1
// ID3;
// // -> Third relation:
// s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*z0^2*s12^5*s12^7*s3i*s12*s12^7*s3i*s12*s12^10*s3*z0^2*s12^5*s12^10*s3*z0^2*s12^5*s12^7*s3i*s12*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2;
// // -> using z0 central and z0^4=1:
// s12^7*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*s12^5*s12^7*s3i*s12*s12^7*s3i*s12*s12^10*s3*s12^5*s12^10*s3*s12^5*s12^7*s3i*s12*s12^7*s3i*s12^2*s12i*s3*s12^5*s12i*s3*s12^5*s12^7*s3i*s12^2;
// // -> Using s12^12=1:
// s12^7*s3i*s12*s3*s12^3*s3*s3i*s12^8*s3i*s12^11*s3*s12^3*s3*s3i*s12^8*s3i*s12*s3*s12^4*s3*s3i*s12^2;
// // -> Using s3*s3i=1
// s12^7*s3i*s12*s3*s12^3*s12^8*s3i*s12^11*s3*s12^3*s12^8*s3i*s12*s3*s12^4*s12^2;
// // -> Using s12^12=1:
// s12^7*s3i*s12*s3*s12^11*s3i*s12^11*s3*s12^11*s3i*s12*s3*s12^6;
// // -> conjugating by s12^6
// s12*s3i*s12*s3*s12^11*s3i*s12^11*s3*s12^11*s3i*s12*s3;
// // -> using s3i=s3*s3
// s12*s3 *s3* s12*s3 *s12^10*s12*s3*s3*s12^10*s12*s3*s12^10*s12*s3*s3*s12*s3;
// // -> using s12*s3=s2^-1=s2*z0
// s2*z0 *s3* s2*z0 *s12^10* s2*z0 *s3*s12^10* s2*z0 *s12^10* s2*z0 *s3* s2*z0;
// // -> using z0 central and z0^4=1:
// s2*s3*s2*s12^10*s2*s3*s12^10*s2*s12^10*s2*s3*s2*z0^2;
// // -> using s3*s2=s12^-1
// s2*s12^9*s2*s3*s12^10*s2*s12^10*s2*z0^2*s12^-1;
// // -> using s3*s2=s12^-1
// s2*s12*s12^8*s2*s3*s12^10*s2*s12*s12^9*s2*s12*s12^10*z0^2;
// // -> using s2*s12=s3i
// s3i*s12^8*s2*s3*s12^10*s3i*s12^9*s3i*s12^10*z0^2;
// // -> invert, and use s12^12=1
// z0^-2*s12^2*s3*s12^3*s3*s12^2*s3i*s2^-1*s12^4*s3*z0^2;
// // -> using z0 central and z0^4=1 and s2^-1=s2*z0
// s12*s12*s3*s12^2*s12*s3*s12*s12*s3*s3*s2*s12^3*s12*s3*z0^3;
// // -> using s12*s3=s2^-1=s2*z0
// s12*s2*z0*s12^2*s2*z0*s12*s2*z0*s3*s2*s12^3*s2*z0*z0^3;
// // -> using z0 central and z0^4=1
// s12*s2*s12*s12*s2*s12*s2*s3*s2*s12*s12^2*s2*z0^3;
// // -> using s2*s12=s3^=1=s3*s3
// s12*s3*s3*s12*s3*s3*s2*s3*s3*s3*s12^2*s2*z0^3;
// // -> using s3^3=1
// s12*s3*s3*s12*s3*s3*s2*s12*s12*s2*z0^3;
// // -> using s3*s2*s12=1
// s12*s3*s3*s12*s3*s12*s2*z0^3;
// // -> using s12*s3=s2^-1=s2*z0
// s2*z0*s3*s2*z0*s12*s2*z0^3;
// // -> using z0 central and z0^4=1
// s2*s3*s2*s12*s2*z0;
// // -> using s3*s2*s12=1
// s2*s2*z0;
// // -> using s2^-1=s2*z0
// ID3;
// 
// // Fourth relation:
// (s12^10*s3*z0^2*s12^5)^12;
// //-> using z0 central and z0^4=1
// (s12^10*s3*s12^5)^12;
// //-> conjugating by s12^5
// (s12^3*s3)^12;
// //-> using s12*s3=s2^-1=s2*z0
// (s12^2*s2*z0)^12;
// //-> using z0 central and z0^4=1
// (s12^2*s2)^12;
// //-> conjugating by s12^-1
// (s12*s2*s12)^12;
// //-> using s2*s12=s3^-1
// (s12*s3i)^12;
// //-> conjugating by s3
// (s3*s12*s3)^12;
// //-> using s12*s3=s2^-1=s2*z0
// (s3*s2*z0)^12;
// //-> using z0 central and z0^4=1
// (s3*s2)^12;
// //-> using s3*s2=s12i
// (s12i)^12;
// //-> using s12^12=1
// ID3;
// 
// // Fifth relation
// s12^10*s3*z0^2*s12^5*s12^10*s3*z0^2*s12^5*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2*s12^7*z0^2*s3i*s12^2*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*z0^2*s12^5*s12^10*s3*z0^2*s12^5*s12i*s3*s12^5*s12^7*z0^2*s3i*s12^2*s12^7*s3i*s12*s12^7*z0^2*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*z0^2*s12^5;
// // -> using z0 central and z0^4=1
// s12^10*s3*s12^5*s12^10*s3*s12^5*s12i*s3*s12^5*s12^7*s3i*s12^2*s12^7*s3i*s12^2*s12^7*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*s12^5*s12^10*s3*s12^5*s12i*s3*s12^5*s12^7*s3i*s12^2*s12^7*s3i*s12*s12^7*s3i*s12^2*s12i*s3*s12^5*s12^10*s3*s12^5;
// // -> using s12^12=1
// s12^10*s3*s12^3*s3*s12^4*s3*s3i*s12^9*s3i*s12^9*s3i*s12*s3*s12^3*s3*s12^3*s3*s12^4*s3*s3i*s12^9*s3i*s12^8*s3i*s12*s3*s12^3*s3*s12^5;
// // -> using s3*s3i=1
// s12^10*s3*s12^3*s3*s12^4*s12^9*s3i*s12^9*s3i*s12*s3*s12^3*s3*s12^3*s3*s12^4*s12^9*s3i*s12^8*s3i*s12*s3*s12^3*s3*s12^5;
// // -> using s12^12=1
// s12^10*s3*s12^3*s3*s12*s3i*s12^9*s3i*s12*s3*s12^3*s3*s12^3*s3*s12*s3i*s12^8*s3i*s12*s3*s12^3*s3*s12^5;
// // -> conjugating by s12^5, writing s3i=s3*s3
// s12^3*s3*s12^3*s3*s12*s3*s3*s12^9*s3*s3*s12*s3*s12^3*s3*s12^3*s3*s12*s3*s3*s12^8*s3*s3*s12*s3*s12^3*s3;
// // -> writing s12^j*s3=s12^(j-1)*s12*s3 if j\ge2
// s12^2*s12*s3*s12^2*s12*s3*s12*s3*s3*s12^8*s12*s3*s3*s12*s3*s12^2*s12*s3*s12^2*s12*s3*s12*s3*s3*s12^7*s12*s3*s3*s12*s3*s12^2*s12*s3;
// // -> using s12*s3=s2^-1=s2*z0
// s12^2*s2*z0*s12^2*s2*z0*s2*z0*s3*s12^8*s2*z0*s3*s2*z0*s12^2*s2*z0*s12^2*s2*z0*s2*z0*s3*s12^7*s2*z0*s3*s2*z0*s12^2*s2*z0;
// // -> using z0 central and z0^4=1
// s12^2*s2*s12^2*s2*s2*s3*s12^8*s2*s3*s2*s12^2*s2*s12^2*s2*s2*s3*s12^7*s2*s3*s2*s12^2*s2*z0^3;
// // -> writing s2*s12^j=s2*s12*s12^(j-1) if j\ge2
// s12*s12*s2*s12*s12*s2*s2*s3*s12^8*s2*s3*s2*s12*s12*s2*s12*s12*s2*s2*s3*s12^7*s2*s3*s2*s12*s12*s2*z0^3;
// // -> using s2*s12=s3^-1=s3*s3
// s12*s12*s3*s3*s12*s2*s2*s3*s12^8*s2*s3*s3*s3*s12*s3*s3*s12*s2*s2*s3*s12^7*s2*s3*s3*s3*s12*s2*z0^3;
// // -> using s3^3=1
// s12*s12*s3*s3*s12*s2*s2*s3*s12^8*s2*s12*s3*s3*s12*s2*s2*s3*s12^7*s2*s12*s2*z0^3;
// // -> using s2*s2=z0^3, which is central
// s12*s12*s3*s3*s12*s3*s12^8*s2*s12*s3*s3*s12*s3*s12^7*s2*s12*s2*z0;
// // -> using s2*s12*s3=1
// s12*s12*s3*s3*s12*s3*s12^7*s12*s3*s12*s3*s12^7*s2*s12*s2*z0;
// // -> using s12*s3=s2^-1=s2*z0
// s12*s2*z0*s3*s2*z0*s12^7*s2*z0*s2*z0*s12^7*s2*s12*s2*z0;
// // -> using z0 central and z0^4=1
// s12*s2*s3*s2*s12^7*s2*s2*s12^7*s2*s12*s2*z0;
// // -> using s2*s2=z0^3, which is central
// s12*s2*s3*s2*s12*s12^6*s12^7*s2*s12*s2;
// // -> using s3*s2*s12=1
// s12*s2*s12^6*s12^7*s2*s12*s2;
// // -> using s12^12=1
// s12*s2*s12*s2*s12*s2;
// // -> conjugating by s12^-1
// s2*s12*s2*s12*s2*s12;
// // -> using s2*s12=s3i
// s3i*s3i*s3i;
// // -> using s3^3=1.

// The presentation of the group $\bar\Gamma_c$:
// *********************************************

// We set
t4:=J^-1*(B*U^-1)^2;
t12:=J;
t2:=(B*U^-1)^2;
zc:=U;

// The following being zero mean that t2,t4,t12 and zc fix $M_c$:
t2[1][2]-c*(c*t2[2][1]-t2[1][1]+t2[2][2]);
t4[1][2]-c*(c*t4[2][1]-t4[1][1]+t4[2][2]);
t12[1][2]-c*(c*t12[2][1]-t12[1][1]+t12[2][2]);
zc[1][2]-c*(c*zc[2][1]-zc[1][1]+zc[2][2]);

// THE FOLLOWING ARE zero:
t2^2 - ID3;
t4^4 - z^4*zc;
t12^12 - ID3;
t2*zc - zc*t2;
t4*zc - zc*t4;
t12*zc - zc*t12;
t12*t4*t2 - ID3;

// This means that there is a homomorphism from the abstract group with
// the presentation stated for $\bar\Gamma_c$ in Section 2.1 to the group $\bar\Gamma_c$.
// See the commented out LateX file for the definition of the group $U_c$ and
// the injective homomorphism $\psi_c:\bar\Gamma_c\to U_c$. As proved in that file,
// the image of $\bar\Gamma_c$ under~$\psi_c$ has index 4 in $U_c$.
// Here we set up that map $\psi_c$:

psic:=function(matvar);
if ((matvar[1][1]-c*matvar[2][1] eq 0) or (matvar[1][3] ne c*matvar[2][3]) 
    or (matvar[1][2] ne c*(c*matvar[2][1]-matvar[1][1]+matvar[2][2])))
 then print "Error - matrix does not stabilize M_c"; return matvar; end if;
tmpmat1:=(1/(matvar[1][1]-c*matvar[2][1]))*matvar;
tmpmat2:=MatsL![1,0,0,0,1,0,0,0,1/Determinant(tmpmat1)]*tmpmat1;
resultmat1:=Mats2L![hLL(tmpmat2[3][3]),r*tmpmat2[2][3],hLL(tmpmat2[2][3]),tmpmat2[3][3]];
return Mats2L![1,0,0,Determinant(tmpmat1)]*resultmat1;
end function;

// Here now are the generators of U_c. As well as D (as above), we have:

S:=Mats2L![z^3*(r+1),r/c,
           1/hLL(c),z^-3*(r+1)];

Q:=Mats2L![r+1,r/c,
           1/hLL(c),r+1];

Fc:=Mats2L![1,0,
            0,-r];

// The following are zero:
psic((1/z^4)*J^7*B*UI*B*U*J^7) - S*D;
psic((1/z^4)*J^9*B*UI*B*U*J^5) - D*S;
psic(J^4*B*UI*B*UI*J^2) - Q;
psic(z*JI)-ZMAT;

// The group $U_c$ is generated by the above matrices D, Q, S and ZMAT, and
// has the presentation
// $\langle d,q,s,z\ |\ d^{12}=z^{12}=1,s^2=(qd^3)^2=(qd^2sd^2)^2=d^6,\ zd=dz, zsz^{-1}=dqd^2, zqz^{-1}=d^{-2}sd^{-1}\rangle$.

// Assuming this, we now explain why $\bar\Gamma_c$ has the presentation of the
// central extension given in the write-up:

FUc<xd,xq,xs,xz>:=FreeGroup(4);
relnlistc:={
xd^12,
xs^2*xd^-6,
(xq*xd^3)^2*xd^-6,
(xq*xd^2*xs*xd^2)^2*xd^-6,
xz^12,
xz*xd*xz^-1*xd^-1,
xz*xq*xz^-1*xd*xs^-1*xd^2,
xz*xs*xz^-1*xd^-2*xq^-1*xd^-1};

Uc,phic:=quo< FUc | relnlistc>;

d:=phic(xd);
q:=phic(xq);
s:=phic(xs);
zabstract:=phic(xz);

testsubgroup:=sub< Uc | s*d,d*s,q,zabstract>;
// print "The index of testsubgroup in Uc is",Index(Uc,testsubgroup); // Output: 4
// print "It is",IsNormal(Uc,testsubgroup),"that testsubgroup is normal in Uc"; // Output: true
// testsubgroup is contained in the image $\psi_c(\bar\Gamma_c)$ of $\bar\Gamma_c$ in~$U_c$,
// and has index~4 in $U_c$. Now $d,d^2,d^3\not\in\psi_c(\bar\Gamma_c)$, since 
// $\zeta^i-1\not\in r\Z[\zeta]$ for $i=1,2,3$, and so the index of~$\psi_c(\bar\Gamma_c)$ 
// in~$\U_c$ is at least~4. We conclude that $\psi_c(\bar\Gamma_c)$ equals testsubgroup.

psicbargammac:=testsubgroup;
print "The index of psicbargammac in Uc is",Index(Uc,psicbargammac); // Output: 4
print "It is",IsNormal(Uc,psicbargammac),"that psicbargammac is normal in Uc"; // Output: true

// R:=Rewrite(Uc,psicbargammac);
// print R;

//Output:

// Finitely presented group R on 3 generators
// Generators as words in group Uc
// R1=Uc.4
// R2=Uc.1*Uc.2*Uc.1^-1
// R3=Uc.3^2*Uc.1^-2
// Relations
// R3^3=Id(R)
// (R3^-1,R1)=Id(R)
// R1^2*R2*R3^-1*R1^-2*R2*R3=Id(R)
// R1^-1*R2*R1*R3*R2*R3*R1*R2*R1^-1*R2^-1=Id(R)
// R1^12=Id(R)

R1:=ZMAT;
R2:=D*Q*D^-1;
R3:=S^2*D^-2;
// The following are zero:
psic(z*JI) - ZMAT;
psic(J^6*B*UI*B*UI)-D*Q*D^-1;
psic(z^(-4)*UI*J^8) - S^2*D^-2;

// In terms of the given generators $t_2=(bu^{-1})^2$, $t4=j^{-1}(bu^{-1})^2$, $t_{12}=j$ 
// and $z_c=u$ of the central extension, we have

psic(z*t12^-1) - R1;
psic(t12^6*t2) - R2;
psic(z^(-4)*zc^-1*t12^8) - R3;

//  We now replace R1, R2 and R3 in the five relations found above by the Rewrite routine
//  by t12^-1, t12^6*t2 and zc^-1*t12^8, respectively, and check that they
//  are consequences of the stated relations amongst t_2, t_4, t_{12} and z_c of the central extension.
//  This means that there is a homomorphism from $\psi_c(\bar\Gamma_c)$ to the central extension
//  mapping R1 to t12^-1, R2 to t12^6*t2 and R3 to zc^-1*t12^8
//  write
t2i:=t2^-1;
t4i:=t4^-1;
t12i:=t12^-1;
zci:=zc^-1;
// Here are the five relations found by the Rewrite routine:
R3^3;
R3^-1*R1*R3*R1^-1;
R1^2*R2*R3^-1*R1^-2*R2*R3;
R1^-1*R2*R1*R3*R2*R3*R1*R2*R1^-1*R2^-1;
R1^12;

// Expressed in terms of t2, t4, t12 and zc, they are:
(zci*t12^8)^3;
t12^4*zc*t12i*(zci*t12^8)*t12;
t12i^2*(t12^6*t2)*t12^4*zc*t12^2*(t12^6*t2)*(zci*t12^8);
t12*(t12^6*t2)*t12i*(zci*t12^8)*(t12^6*t2)*(zci*t12^8)*t12i*(t12^6*t2)*t12*t2i*t12^6;
t12i^12;

// // First relation
// (zc^-1*t12^8)^3;
// // -> using zci=zc^-1 is central
// (zc^-1)^3*t12^24;
// // Using t12^12=1=zc^3:
// ID3;
// 
// // Second relation:
// t12^4*zc*t12i*zci*t12^8*t12;
// //-> using zc and zci=zc^-1 are central
// t12^4*t12i*t12^8*t12;
// // -> using t12^12=1
// ID3;
// 
// // Third relation:
// t12i^2*t12^6*t2*t12^4*zc*t12^2*t12^6*t2*zci*t12^8;
// //-> using zc and zci=zc^-1 are central
// t12i^2*t12^6*t2*t12^4*t12^2*t12^6*t2*t12^8;
// // -> using t12^12=1
// t12^4*t2*t2*t12^8;
// // -> using t2^2=1
// t12^4*t12^8;
// // -> using t12^12=1
// ID3;
// 
// // Fourth relation:
// t12*t12^6*t2*t12i*zci*t12^8*t12^6*t2*zci*t12^8*t12i*t12^6*t2*t12*t2i*t12^6;
// // -> using zc and zci=zc^-1 are central
// t12*t12^6*t2*t12i*t12^8*t12^6*t2*t12^8*t12i*t12^6*t2*t12*t2i*t12^6*zci^2;
// // -> using t12^12=zc^3=t2^2=1
// t12^7*t2*t12*t2*t12*t2*t12*t2*t12^6*zc;
// // -> using t2*t12=t4i
// t12^7*t4i*t4i*t4i*t4i*t12^5*zc;
// // -> using t4^4=zc
// t12^7*zci*t12^5*zc;
// // -> using zc and zci=zc^-1 are central and t12^12=1
// ID3;
// 
// // Fifth relation:
// t12i^12;
// // -> using t12^12=1
// ID3;

// end of calculations for Section 2.1.
// ************************************
//
// Calculations for Proposition 2:
// *******************************
// Now let's look at the subgroup $\bar\Gamma_0$ consisting of the
// $g\in\bar\Gamma$ such that $g(M_0)=M_0$. The image under $\rho'$ 
// (see calculations for Section 1.6) of such an element has the form
// (*,0,0),
// (0,*,*),     (1)
// (0,*,*)
// So reducing mod scalars, the image is contained in $U(2,\F_9)$,
// which has order 96. Let $\bar\Gamma^1_0$ denote the subgroup
// of $\bar\Gamma_0$ consisting of the elements having a matrix
// representative of determinant 1. Since $-bj$, $\zeta^{-1}v$ and $-j^3$ are
// all in $\bar\Gamma^1_0$, we see that already the image of $\bar\Gamma^1_0$
// is equal to $U(2,\F_9)$.
testgp:=sub< GL3F9 | rhodBD*rhodJD,rhodVD,rhodJD^3>;
#testgp; // Output: 384 = 96 times 4.
// Notice that the image under $\rho'$ of $\Pi_0=\bar\Gamma_0\cap\Pi=\bar\Gamma^1_0\cap\Pi$
// must be trivial, since the only element of $\langle R,M\rangle$ of the form (1)
// is the trivial element.
// Write $\bar\Gamma^1_0$ as a union of cosets $\Pi_0t$, $t\in T$, say.
// Then the images $\rho'(t)$, $t\in T$, are distinct, since $\rho'(t)=\rho'(t')$
// implies that $t't^{-1}$ is in the kernel of~$\rho'$ and so in the preimage of $\langle R,M\rangle$.
// Hence $|T|=96$. That is $[\bar\Gamma^1_0,\Pi_0]=96$. Since $[\bar\Gamma_0,\bar\Gamma^1_0]=3$,
// we have checked that $[\bar\Gamma_0,\Pi_0]=288$.

// The following matrices are in~$\Pi_0$. Because of the scalar factors,
// they all have the form
// (1,0,0),
// (0,a,b),
// (0,c,d)

g1:=z^5*A3I*A3I*A3I*A1I*A2*A1;
g3:=(1/z^4)*A2*A1I*A1I*A3I*A3I*A3I*A1I;
g5:=z^3*J^4*A2*A1*J^8*A2I*A3*A3*A3*A1*A1;
g7:=J^4*A1I*A2I*J^4*A2*A1*J^4;
g2:=J^4*g1*J^8;
g4:=J^4*g3*J^8;
g6:=J^4*g5*J^8;
g8:=J^4*g7*J^8;

g1I:=g1^-1;
g2I:=g2^-1;
g3I:=g3^-1;
g4I:=g4^-1;
g5I:=g5^-1;
g6I:=g6^-1;
g7I:=g7^-1;
g8I:=g8^-1;

// The following equals the identity matrix:
g1*g2*g3*g4*g5*g6*g7*g8*g1I*g3I*g5I*g7I*g2I*g4I*g6I*g8I;

// We can express the g_i's as words in the generators s2,s3, s12 and z0
// of $\bar\Gamma_0$:
// The following are zero:

g1 - (1/z)*s12^4*s2*s12^4*s2*s12^4*z0;
g2 - (1/z)*s12^8*s2*s12^4*s2*z0;
g3 - (1/z)*s12^11*s2*s12^4*s2*s12^9*z0;
g4 - (1/z)*s12^3*s2*s12^4*s2*s12^5*z0;
g5 - (1/z)*s12^6*s2*s12^4*s2*s12^2*z0;
g6 - (1/z)*s12^10*s2*s12^4*s2*s12^10*z0;
g7 - (1/z)*s12*s2*s12^4*s2*s12^7*z0;
g8 - (1/z)*s12^5*s2*s12^4*s2*s12^3*z0;

g1I -  (1/z)*s12^8*s2*s12^8*s2*s12^8*z0;
g2I -  (1/z)*s2*s12^8*s2*s12^4*z0;
g3I -  (1/z)*s12^3*s2*s12^8*s2*s12*z0;
g4I -  (1/z)*s12^7*s2*s12^8*s2*s12^9*z0;
g5I - (1/z)*s12^10*s2*s12^8*s2*s12^6*z0;
g6I - (1/z)*s12^2*s2*s12^8*s2*s12^2*z0;
g7I -  (1/z)*s12^5*s2*s12^8*s2*s12^11*z0;
g8I -  (1/z)*s12^9*s2*s12^8*s2*s12^7*z0;

// We can then use our presentation of $\bar\Gamma_0$ to check that
// $\langle g_1,\ldots,g_8\rangle$ has index 288 in $\bar\Gamma_0$,
// and so equals $\Pi_0$. See the other magma file.

// Replacing $g_1,\ldots,g_8$ by generators satisfying the canonical relation:
// Set
D1:=g1*g2*g3*g4*g5*g6*g7;
D2:=g1*g2*g3*g4;
D3:=g1;
D4:=g3I;

E1:=g8*g1I*g3I*g5I;    
E2:=g5*g6*g2I; 
E3:=g2*g3*g6I;
E4:=g6; 

// The following are words in the Di's and Ei's and so words in g1,...,g8:
U1:=D1;                            V1:=E1;
U2:=E1*D2*E1^-1;                   V2:=E1*E2*E1^-1;
U3:=E1*E2*D3*E2^-1*E1^-1;          V3:=E1*E2*E3*E2^-1*E1^-1;
U4:=E1*E2*E3*D4*E3^-1*E2^-1*E1^-1; V4:=E1*E2*E3*E4*E3^-1*E2^-1*E1^-1;

// The following is zero:
ID3 - (U1*V1*U1^-1*V1^-1)*(U2*V2*U2^-1*V2^-1)*(U3*V3*U3^-1*V3^-1)*(U4*V4*U4^-1*V4^-1);   // ID3 is a product of four commutators

// Checking that gi's are words in the Ui's and Vi's:

U1I:=U1^-1;
U2I:=U2^-1;
U3I:=U3^-1;
U4I:=U4^-1;

V1I:=V1^-1;
V2I:=V2^-1;
V3I:=V3^-1;
V4I:=V4^-1;

// The following are zero:
g1 - V1I*V2I*U3*V2*V1;
g2 - V1I*V2I*V4*U4*V3*V2*V1;
g3 - V1I*V2I*V3I*U4I*V3*V2*V1;
g4 - V1I*V2I*V3I*V4I*U3I*V2*U2*V1;
g5 - V1I*V4*U4*V4I*V3*V2*V1;
g6 - V1I*V2I*V3I*V4*V3*V2*V1;
g7 - V1I*V2I*V3I*U4I*V4I*U2I*V1*U1;
g8 - V4*U4*V4I*U4I*V3*U3*V2*V1;

// End of calculations for Proposition 2.
// **************************************

// Calculations for Proposition 3.
// *******************************
// (a) Firstly, see the other Magma file for the check that the elements $b^\mu k$, $\mu=0,1,-1$, $k\in K$,
//     form a set of coset representatives for the cosets $\Pi g$ of $\Pi$ in $\bar\Gamma$.

// // The following routine finds elements $\pi$ of~$\Pi$ of minimal 
// // length (if there are any of length at most 6) such that $\pi(b^\mu(M_\alpha))=M_\beta$. 
// // Here $\beta\in\{0,1,\infty\}$, $\alpha\in\{0,1,-1,i,-i,\infty\}$ and $\mu\in\{0,1,-1\}$.
// genlist:=[MatsL | ID3,A1,A1I,A2,A2I,A3,A3I];
// genletters:=[[],["A1"],["A1I"],["A2"],["A2I"],["A3"],["A3I"]];
// possiblefollowingletters:=[
// [1,2,3,4,5,6,7],
// [2,4,5,6,7],
// [3,4,5,6,7],
// [2,3,4,6,7],
// [2,3,5,6,7],
// [2,3,4,5,6],
// [2,3,4,5,7]];
// 
// // Here are the elements listed in the second table of Lemma 2:
// kzero:=ID3;
// kplusi:=U*J;
// kminus1:=V*U*J;
// kminusi:=V^2*U*J;
// kplus1:=V^3*U*J;
// kinfty:=UI*V^2*U*J^6;
// 
// kalphalist:=[MatsL | kzero,kplus1,kminus1,kplusi,kminusi,kinfty];
// alphalist:=["0","1","-1","i","-i","infty"];
// print "\nFinding pi in Pi and beta in {0,1,infty} such that pi(b^mu(M_alpha))=M_beta.\n";
// for mu in [0,1,-1] do
//  for alpha in [1,2,3,4,5,6] do
//   for beta in [1,2,6] do
//    firstlist:=[MatsL | kalphalist[beta]^-1*ID3,kalphalist[beta]^-1*A1,kalphalist[beta]^-1*A1I,kalphalist[beta]^-1*A2,kalphalist[beta]^-1*A2I,kalphalist[beta]^-1*A3,kalphalist[beta]^-1*A3I];
//    lastlist:=[MatsL | B^mu*kalphalist[alpha],A1*B^mu*kalphalist[alpha],A1I*B^mu*kalphalist[alpha],A2*B^mu*kalphalist[alpha],A2I*B^mu*kalphalist[alpha],A3*B^mu*kalphalist[alpha],A3I*B^mu*kalphalist[alpha]];
//    for i1 in [1 .. 7] do
//     for i2 in possiblefollowingletters[i1] do
//      tmpmat1:=firstlist[i1]*genlist[i2];
//      for i3 in possiblefollowingletters[i2] do
//       tmpmat2:=tmpmat1*genlist[i3];
//       for i4 in possiblefollowingletters[i3] do
//        tmpmat3:=tmpmat2*genlist[i4];
//        for i5 in possiblefollowingletters[i4] do
//         tmpmat4:=tmpmat3*genlist[i5];
//         for i6 in possiblefollowingletters[i5] do
//          MM:=tmpmat4*lastlist[i6];
//          if ((MM[1][2] eq 0) and (MM[1][3] eq 0)) then // Test whether $m$ is in $\bar\Gamma_0$ for $m=k_\beta^-1\pi b^\mu k_\alpha$
//           print "When mu = ",mu," and alpha=",alphalist[alpha],", the answer is:";
//           print "beta=",alphalist[beta]," and pi=",genletters[i1] cat genletters[i2] cat genletters[i3] cat genletters[i4] cat genletters[i5] cat genletters[i6];
//           break beta;
//          end if;
//         end for;
//        end for;
//       end for;
//      end for;
//     end for;
//    end for;
//   end for;
//  end for;  
// end for;
// 
// // Output:
// 
// Finding pi in Pi and beta in {0,1,infty} such that pi(b^mu(M_alpha))=M_beta.
// 
// When mu =  0  and alpha= 0 , the answer is:
// beta= 0  and pi= []
// When mu =  0  and alpha= 1 , the answer is:
// beta= 1  and pi= []
// When mu =  0  and alpha= -1 , the answer is:
// beta= 0  and pi= [ A1I, A2I ]
// When mu =  0  and alpha= i , the answer is:
// beta= 0  and pi= [ A1I, A2I, A1 ]
// When mu =  0  and alpha= -i , the answer is:
// beta= infty  and pi= [ A2I ]
// When mu =  0  and alpha= infty , the answer is:
// beta= infty  and pi= []
// 
// When mu =  1  and alpha= 0 , the answer is:
// beta= 0  and pi= []
// When mu =  1  and alpha= 1 , the answer is:
// beta= 1  and pi= [ A2I, A2I ]
// When mu =  1  and alpha= -1 , the answer is:
// beta= 1  and pi= [ A3, A3, A3, A1, A1, A2I ]
// When mu =  1  and alpha= i , the answer is:
// beta= 1  and pi= [ A1I, A3I, A3I, A3I ]
// When mu =  1  and alpha= -i , the answer is:
// beta= 1  and pi= [ A2I ]
// When mu =  1  and alpha= infty , the answer is:
// beta= infty  and pi= [ A2I, A1, A2I ]
// 
// When mu =  -1  and alpha= 0 , the answer is:
// beta= 0  and pi= []
// When mu =  -1  and alpha= 1 , the answer is:
// beta= infty  and pi= [ A3I, A1I, A2I ]
// When mu =  -1  and alpha= -1 , the answer is:
// beta= infty  and pi= [ A3I, A1I, A2I, A2I ]
// When mu =  -1  and alpha= i , the answer is:
// beta= 1  and pi= [ A3I, A3I ]
// When mu =  -1  and alpha= -i , the answer is:
// beta= infty  and pi= [ A2I, A2I ]
// When mu =  -1  and alpha= infty , the answer is:
// beta= 0  and pi= [ A1I, A1I, A3I, A3I, A3I ]

// Magma is not needed in proving Proposition 3(b).
// See the other Magma file for the checks needed in Proposition 3(c).
// Magma is not needed in proving Proposition 3(d).
//
// End of calculations for Proposition 3.
// **************************************

// Calculations for the end of Section 2.3.
// ****************************************

// Here we express the 8 generators $k_1g_jk_1^{-1}$ of $\Pi_1$ 
// and the 8 generators $k_\infty g_jk_\infty^{-1}$ of $\Pi_\infty$
// in terms of the generators $a_1$, $a_2$ and $a_3$ of $\Pi$:

// The following are zero:
kplus1*g1*kplus1^-1 - (z^2)*A2I*A2I*A2I*A3*A3*A3*A1*A2*A1*A3*A1*A2;
kplus1*g3*kplus1^-1 - (z^3)*J^4*(A2I*A2I*A1I*A3I*A1*A2*A2*A1I*A2I*A1)*J^8;
kplus1*g5*kplus1^-1 - (z^5)*J^8*(A1I*A3I*A1*A2*A2*A1I*A2I*A1*A3*A3)*J^4;
kplus1*g7*kplus1^-1 - (-z)*J^4*(A1I*A1I*A3I*A3I*A3I*A1I*A3I)*J^8;

// Recall that the following are also zero:
kplus1*g2*kplus1^-1 - J^4*(kplus1*g1*kplus1^-1)*J^8;
kplus1*g4*kplus1^-1 - J^4*(kplus1*g3*kplus1^-1)*J^8;
kplus1*g6*kplus1^-1 - J^4*(kplus1*g5*kplus1^-1)*J^8;
kplus1*g8*kplus1^-1 - J^4*(kplus1*g7*kplus1^-1)*J^8;

// The following are zero:
kinfty*g1*kinfty^-1 - (1/z^4)*J^4*A1I*A3I*A3I*A1I*J^8*A1I*A2I;
kinfty*g3*kinfty^-1 - (1/z^2)*J^8*A3*A1*A2*A1I*A2I*J^4;
kinfty*g5*kinfty^-1 - (1/z)*J^8*A2I*A3I*J^4;
kinfty*g7*kinfty^-1 - (1/z^2)*J^4*A1*A3*A1I*A3I*A3I*J^8;

// Recall that the following are also zero:
kinfty*g2*kinfty^-1 - J^4*(kinfty*g1*kinfty^-1)*J^8;
kinfty*g4*kinfty^-1 - J^4*(kinfty*g3*kinfty^-1)*J^8;
kinfty*g6*kinfty^-1 - J^4*(kinfty*g5*kinfty^-1)*J^8;
kinfty*g8*kinfty^-1 - J^4*(kinfty*g7*kinfty^-1)*J^8;


// Calculations for Proposition 4:
// ******************************

// Here are the 20 elements of $\Pi_c$ listed in Proposition 4:

Gc1:=J^8*A1I*A2*A1*A3*A1I*J^4*A2*A1;
Gc3:=z*J^4*A2*A1*A2I*A2I*A1I*A3*J^4*A3*A3*A3*J^4;
Gc5:=z^4*J^8*A1I*J^4*A2*A1*J^4*A3*A2I*A1*A3*A1I*J^8;
Gc7:=(1/z^5)*J^8*A2*A1*J^4*A3I*J^4*A2*A1I*A2I*A3I*A3I*A3I*J^8;
Gc9:=z^4*J^8*A1I*A2I*A2I*A1I*A3I*J^8*A1I*A2I*J^8;
Gc12:=(-1)*A2I*A1*A3*A1I*A3I*J^4*A3*A1*A2*A2*A1I*A2I*J^8;
Gc15:=z*J^4*A1*J^4*A2*A3*A1I*J^4;
Gc17:=(1/z^2)*J^8*A1I*A1I*A2I*J^4*A3*A1*A2*A1;
Gc19:=(1/z)*A2I*A1*A3*A1I*A3I*A3I*J^4*A1*A2*J^4*A1I*A2I*J^4;
Gc2:=J^4*Gc1*J^8;
Gc4:=J^4*Gc3*J^8;
Gc6:=J^4*Gc5*J^8;
Gc8:=J^4*Gc7*J^8;
Gc10:=J^4*Gc9*J^8;
Gc11:=J^4*Gc10*J^8;
Gc13:=J^4*Gc12*J^8;
Gc14:=J^4*Gc13*J^8;
Gc16:=J^4*Gc15*J^8;
Gc18:=J^4*Gc17*J^8;
Gc20:=J^4*Gc19*J^8;

Gc1I:=Gc1^-1;
Gc2I:=Gc2^-1;
Gc3I:=Gc3^-1;
Gc4I:=Gc4^-1;
Gc5I:=Gc5^-1;
Gc6I:=Gc6^-1;
Gc7I:=Gc7^-1;
Gc8I:=Gc8^-1;
Gc9I:=Gc9^-1;
Gc10I:=Gc10^-1;
Gc11I:=Gc11^-1;
Gc12I:=Gc12^-1;
Gc13I:=Gc13^-1;
Gc14I:=Gc14^-1;
Gc15I:=Gc15^-1;
Gc16I:=Gc16^-1;
Gc17I:=Gc17^-1;
Gc18I:=Gc18^-1;
Gc19I:=Gc19^-1;
Gc20I:=Gc20^-1;


// Checking that the Gci are in $\bar\Gamma_c$. The following are zero:

Gc1[1][3]-c*Gc1[2][3]; Gc1[1][2] - c*(c*Gc1[2][1]-Gc1[1][1]+Gc1[2][2]);
Gc2[1][3]-c*Gc2[2][3]; Gc2[1][2] - c*(c*Gc2[2][1]-Gc2[1][1]+Gc2[2][2]);
Gc3[1][3]-c*Gc3[2][3]; Gc3[1][2] - c*(c*Gc3[2][1]-Gc3[1][1]+Gc3[2][2]);
Gc4[1][3]-c*Gc4[2][3]; Gc4[1][2] - c*(c*Gc4[2][1]-Gc4[1][1]+Gc4[2][2]);
Gc5[1][3]-c*Gc5[2][3]; Gc5[1][2] - c*(c*Gc5[2][1]-Gc5[1][1]+Gc5[2][2]);
Gc6[1][3]-c*Gc6[2][3]; Gc6[1][2] - c*(c*Gc6[2][1]-Gc6[1][1]+Gc6[2][2]);
Gc7[1][3]-c*Gc7[2][3]; Gc7[1][2] - c*(c*Gc7[2][1]-Gc7[1][1]+Gc7[2][2]);
Gc8[1][3]-c*Gc8[2][3]; Gc8[1][2] - c*(c*Gc8[2][1]-Gc8[1][1]+Gc8[2][2]);
Gc9[1][3]-c*Gc9[2][3]; Gc9[1][2] - c*(c*Gc9[2][1]-Gc9[1][1]+Gc9[2][2]);
Gc10[1][3]-c*Gc10[2][3]; Gc10[1][2] - c*(c*Gc10[2][1]-Gc10[1][1]+Gc10[2][2]);
Gc11[1][3]-c*Gc11[2][3]; Gc11[1][2] - c*(c*Gc11[2][1]-Gc11[1][1]+Gc11[2][2]);
Gc12[1][3]-c*Gc12[2][3]; Gc12[1][2] - c*(c*Gc12[2][1]-Gc12[1][1]+Gc12[2][2]);
Gc13[1][3]-c*Gc13[2][3]; Gc13[1][2] - c*(c*Gc13[2][1]-Gc13[1][1]+Gc13[2][2]);
Gc14[1][3]-c*Gc14[2][3]; Gc14[1][2] - c*(c*Gc14[2][1]-Gc14[1][1]+Gc14[2][2]);
Gc15[1][3]-c*Gc15[2][3]; Gc15[1][2] - c*(c*Gc15[2][1]-Gc15[1][1]+Gc15[2][2]);
Gc16[1][3]-c*Gc16[2][3]; Gc16[1][2] - c*(c*Gc16[2][1]-Gc16[1][1]+Gc16[2][2]);
Gc17[1][3]-c*Gc17[2][3]; Gc17[1][2] - c*(c*Gc17[2][1]-Gc17[1][1]+Gc17[2][2]);
Gc18[1][3]-c*Gc18[2][3]; Gc18[1][2] - c*(c*Gc18[2][1]-Gc18[1][1]+Gc18[2][2]);
Gc19[1][3]-c*Gc19[2][3]; Gc19[1][2] - c*(c*Gc19[2][1]-Gc19[1][1]+Gc19[2][2]);
Gc20[1][3]-c*Gc20[2][3]; Gc20[1][2] - c*(c*Gc20[2][1]-Gc20[1][1]+Gc20[2][2]);
print "Finished checking that the Gc_i's are in barGamma_c.";

// The following is the identity matrix:
Gc4*Gc14^-1*Gc2^-1*Gc17^-1*Gc9*Gc19*Gc20*Gc14*Gc7^-1*Gc10^-1*Gc5^-1*Gc16^-1*Gc3^-1*Gc12^-1*Gc1*Gc2*Gc18^-1*Gc10*Gc19^-1
 *Gc12*Gc8^-1*Gc11^-1*Gc6^-1 *Gc15*Gc16*Gc4^-1*Gc13^-1*Gc1^-1*Gc17*Gc18*Gc11*Gc20^-1*Gc13*Gc7*Gc8*Gc9^-1*Gc5*Gc6*Gc15^-1*Gc3;

// Checking the action of the outer automorphism of $\Pi$ on~$\Pi_c$.
// The following are zero:
J^4*Gc1*J^8 - Gc2;
J^4*Gc2*J^8 - Gc2^-1*Gc1^-1;
J^4*Gc3*J^8 - Gc4;
J^4*Gc4*J^8 - Gc4^-1*Gc3^-1;
J^4*Gc5*J^8 - Gc6;
J^4*Gc6*J^8 - Gc6^-1*Gc5^-1;
J^4*Gc7*J^8 - Gc8;
J^4*Gc8*J^8 - Gc8^-1*Gc7^-1;
J^4*Gc9*J^8 - Gc10;
J^4*Gc10*J^8 - Gc11;
J^4*Gc11*J^8 - Gc9;
J^4*Gc12*J^8 - Gc13;
J^4*Gc13*J^8 - Gc14;
J^4*Gc14*J^8 - Gc12;
J^4*Gc15*J^8 - Gc16;
J^4*Gc16*J^8 - Gc16^-1*Gc15^-1;
J^4*Gc17*J^8 - Gc18;
J^4*Gc18*J^8 - Gc18^-1*Gc17^-1;
J^4*Gc19*J^8 - Gc20;
J^4*Gc20*J^8 - Gc20^-1*Gc19^-1;

// Here we express each of these 20 generators of $\Pi_c$
// in terms of the four generators of $\bar\Gamma_c$ given
// in Section 2.1.

// The following are zero:

Gc1 - t12^6*t2*t12^4*t2*t12^2;
Gc2 - t12^10*t2*t12^4*t2*t12^10;
Gc3 - t12^8*t2*t12^4*t2;
Gc4 - t2*t12^4*t2*t12^8;
Gc5 - t12^5*t2*t12^4*t2*t12^3;
Gc6 - t12^9*t2*t12^4*t2*t12^11;
Gc7 - t12^11*t2*t12^4*t2*t12^9;
Gc8 - t12^3*t2*t12^4*t2*t12^5;
Gc9 - t12^5*t2*t12^10*t2*t12^10*t2*t12^5;
Gc10 - t12^9*t2*t12^10*t2*t12^10*t2*t12;
Gc11 - t12*t2*t12^10*t2*t12^10*t2*t12^9;
Gc12 - t12^6*t2*t12^10*t2*t12^10*t2*t12^4;
Gc13 - t12^10*t2*t12^10*t2*t12^10*t2;
Gc14 - t12^2*t2*t12^10*t2*t12^10*t2*t12^8;
Gc15 - (1/z^4)*t12^8*t2*t12^11*t2*t12^3*t2*t12^11*t2*t12^11*zc^2;
Gc16 - (1/z^4)*t2*t12^11*t2*t12^3*t2*t12^11*t2*t12^7*zc^2;
Gc17 - (1/z^4)*t12^5*t2*t12^11*t2*t12^3*t2*t12^11*t2*t12^2*zc^2;
Gc18 - (1/z^4)*t12^9*t2*t12^11*t2*t12^3*t2*t12^11*t2*t12^10*zc^2;
Gc19 - (1/z^4)*t12^6*t2*t12^11*t2*t12^3*t2*t12^11*t2*t12*zc^2;
Gc20 - (1/z^4)*t12^10*t2*t12^11*t2*t12^3*t2*t12^11*t2*t12^9*zc^2;

Gc1I - t12^10*t2*t12^8*t2*t12^6;
Gc2I - t12^2*t2*t12^8*t2*t12^2;
Gc3I - t2*t12^8*t2*t12^4;
Gc4I - t12^4*t2*t12^8*t2;
Gc5I - t12^9*t2*t12^8*t2*t12^7;
Gc6I - t12*t2*t12^8*t2*t12^3;
Gc7I - t12^3*t2*t12^8*t2*t12;
Gc8I - t12^7*t2*t12^8*t2*t12^9;
Gc9I - t12^7*t2*t12^2*t2*t12^2*t2*t12^7;
Gc10I - t12^11*t2*t12^2*t2*t12^2*t2*t12^3;
Gc11I - t12^3*t2*t12^2*t2*t12^2*t2*t12^11;
Gc12I - t12^8*t2*t12^2*t2*t12^2*t2*t12^6;
Gc13I - t2*t12^2*t2*t12^2*t2*t12^2;
Gc14I - t12^4*t2*t12^2*t2*t12^2*t2*t12^10;
Gc15I - z^4*t12*t2*t12*t2*t12^9*t2*t12*t2*t12^4*zc;
Gc16I - z^4*t12^5*t2*t12*t2*t12^9*t2*t12*t2*zc;
Gc17I - z^4*t12^10*t2*t12*t2*t12^9*t2*t12*t2*t12^7*zc;
Gc18I - z^4*t12^2*t2*t12*t2*t12^9*t2*t12*t2*t12^3*zc;
Gc19I - z^4*t12^11*t2*t12*t2*t12^9*t2*t12*t2*t12^6*zc;
Gc20I - z^4*t12^3*t2*t12*t2*t12^9*t2*t12*t2*t12^2*zc;

print "Expressing the Di and Ei's in terms of the Gci's:";
D1:=Gc4*Gc14I*Gc2I*Gc17I*Gc9*Gc19*Gc20*Gc14*Gc7I*Gc10I*Gc5I*Gc16I;
D2:=Gc4*Gc14I*Gc2I*Gc17I*Gc9*Gc19*Gc20*Gc14*Gc7I*Gc10I*Gc5I*Gc4I*Gc13I*Gc1I*Gc17*Gc18*Gc11*Gc20I*Gc13*Gc7*Gc8*Gc9I*Gc5*Gc6;
D3:=Gc4*Gc14I*Gc2I*Gc17I*Gc9*Gc19*Gc20*Gc14*Gc7I*Gc10I*Gc5I*Gc4I*Gc13I*Gc1I*Gc17*Gc18;
D4:=Gc4*Gc14I;
D5:=Gc4*Gc7I*Gc10I*Gc5I*Gc4I*Gc13I*Gc1I*Gc17*Gc10*Gc19I*Gc12*Gc8I;
D6:=Gc4;
D7:=Gc13I*Gc1I*Gc17*Gc10*Gc19I*Gc12*Gc9I;
D8:=Gc13I*Gc1I;
D9:=Gc13I*Gc10*Gc19I;
D10:=Gc13I;

E1:=Gc3I*Gc12I*Gc1*Gc2*Gc18I*Gc10*Gc19I*Gc12*Gc8I*Gc11I*Gc6I*Gc15;
E2:=Gc15I*Gc12I*Gc1*Gc2*Gc18I*Gc10*Gc19I*Gc12*Gc8I*Gc11I;
E3:=Gc11*Gc20I*Gc13*Gc7*Gc8*Gc9I*Gc5*Gc12I*Gc1*Gc2;
E4:=Gc2I*Gc17I*Gc9*Gc19*Gc20;
E5:=Gc20I*Gc13*Gc7;
E6:=Gc7I*Gc10I*Gc5I;
E7:=Gc5*Gc12I*Gc1*Gc17I;
E8:=Gc17*Gc10*Gc19I*Gc12*Gc19*Gc13*Gc10I*Gc12I;
E9:=Gc12;
E10:=Gc10;

print "Expressing the Ui and Vi's in terms of the Di's and Ei's:";
U1:=D1;
V1:=E1;

U2:=E1*D2*E1^-1;
V2:=E1*E2*E1^-1;

U3:=E1*E2*D3*E2^-1*E1^-1;
V3:=E1*E2*E3*E2^-1*E1^-1;

U4:=E1*E2*E3*D4*E3^-1*E2^-1*E1^-1;
V4:=E1*E2*E3*E4*E3^-1*E2^-1*E1^-1;

U5:=E1*E2*E3*E4*D5*E4^-1*E3^-1*E2^-1*E1^-1;
V5:=E1*E2*E3*E4*E5*E4^-1*E3^-1*E2^-1*E1^-1;

U6:=E1*E2*E3*E4*E5*D6*E5^-1*E4^-1*E3^-1*E2^-1*E1^-1;
V6:=E1*E2*E3*E4*E5*E6*E5^-1*E4^-1*E3^-1*E2^-1*E1^-1;

U7:=E1*E2*E3*E4*E5*E6*D7*E6^-1*E5^-1*E4^-1*E3^-1*E2^-1*E1^-1;
V7:=E1*E2*E3*E4*E5*E6*E7*E6^-1*E5^-1*E4^-1*E3^-1*E2^-1*E1^-1;

U8:=E1*E2*E3*E4*E5*E6*E7*D8*E7^-1*E6^-1*E5^-1*E4^-1*E3^-1*E2^-1*E1^-1;
V8:=E1*E2*E3*E4*E5*E6*E7*E8*E7^-1*E6^-1*E5^-1*E4^-1*E3^-1*E2^-1*E1^-1;

U9:=E1*E2*E3*E4*E5*E6*E7*E8*D9*E8^-1*E7^-1*E6^-1*E5^-1*E4^-1*E3^-1*E2^-1*E1^-1;
V9:=E1*E2*E3*E4*E5*E6*E7*E8*E9*E8^-1*E7^-1*E6^-1*E5^-1*E4^-1*E3^-1*E2^-1*E1^-1;

U10:=E1*E2*E3*E4*E5*E6*E7*E8*E9*D10*E9^-1*E8^-1*E7^-1*E6^-1*E5^-1*E4^-1*E3^-1*E2^-1*E1^-1;
V10:=E1*E2*E3*E4*E5*E6*E7*E8*E9*E10*E9^-1*E8^-1*E7^-1*E6^-1*E5^-1*E4^-1*E3^-1*E2^-1*E1^-1;

U1I:=U1^-1;
U2I:=U2^-1;
U3I:=U3^-1;
U4I:=U4^-1;
U5I:=U5^-1;
U6I:=U6^-1;
U7I:=U7^-1;
U8I:=U8^-1;
U9I:=U9^-1;
U10I:=U10^-1;

V1I:=V1^-1;
V2I:=V2^-1;
V3I:=V3^-1;
V4I:=V4^-1;
V5I:=V5^-1;
V6I:=V6^-1;
V7I:=V7^-1;
V8I:=V8^-1;
V9I:=V9^-1;
V10I:=V10^-1;

print "The following is the identity matrix:";
(U1*V1*U1I*V1I)*(U2*V2*U2I*V2I)*(U3*V3*U3I*V3I)*(U4*V4*U4I*V4I)*(U5*V5*U5I*V5I)*(U6*V6*U6I*V6I)*(U7*V7*U7I*V7I) *(U8*V8*U8I*V8I)*(U9*V9*U9I*V9I)*(U10*V10*U10I*V10I);

print "Checking expressions for the Gci's in terms of the Ui's and Vi's:";
// The following are zero:

Gc1 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*U8I*V8I*V9I*U10*V9*V8*V7*V6*V5*V4*V3*V2*V1;
Gc1I - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*U10I*V9*V8*U8*V7*V6*V5*V4*V3*V2*V1;

Gc2 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*U10I*V9*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V7*U7I*V7I*U8*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V3*V2*V1;
Gc2I - V1I*V2I*V3I*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I*U8I*V7*U7*V7I*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V9I*U10*V9*V8*V7*V6*V5*V4*V3*V2*V1;

Gc3 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V7*U7I*V7I*U8*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V3*U3I*V3I*U4*V4
      *U4I*V4I*U5*V5*U5I*V5I*U6*V6*U6I*V6I*U7*V7*U7I*V7I*U8*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V2*U2I*V2I*U3*V3*U3I*V3I*U4*V4*U4I*V4I*U5*V5*U5I*V5I*U6*V6
        *U6I*V6I*U7*V7*U7I*V7I*U8*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I;
Gc3I - V10*U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I*U8I*V7*U7*V7I*U7I*V6*U6*V6I*U6I*V5*U5*V5I*U5I*V4*U4*V4I*U4I*V3*U3*V3I*U3I*V2*U2*V2I*V10*U10*V10I*U10I*V9*U9*V9I
        *U9I*V8*U8*V8I*U8I*V7*U7*V7I*U7I*V6*U6*V6I*U6I*V5*U5*V5I*U5I*V4*U4*V4I*U4I*V3*U3*V3I*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I*U8I*V7*U7*V7I*V10
         *U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I*V9*V8*V7*V6*V5*V4*V3*V2*V1;

Gc4 - V1I*V2I*V3I*V4I*V5I*U6*V5*V4*V3*V2*V1;
Gc4I - V1I*V2I*V3I*V4I*V5I*U6I*V5*V4*V3*V2*V1;

Gc5 - V1I*V2I*V3I*V4I*V5I*V6I*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I*V9*V8*V7*V6*V5*V4*V3*V2*V1;
Gc5I - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V6*V5*V4*V3*V2*V1;

Gc6 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V7*U7I*V7I*U8*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*U3I*V2*U2*V1;
Gc6I - V1I*U2I*V2I*U3*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I*U8I*V7*U7*V7I*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I*V9*V8*V7*V6*V5*V4*V3*V2*V1;

Gc7 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*V10I*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V5*V4*V3*V2*V1;
Gc7I - V1I*V2I*V3I*V4I*V5I*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I*V10*V9*V8*V7*V6*V5*V4*V3*V2*V1;

Gc8 -  V1I*V2I*V3I*V4I*U5I*V5I*U6*V6*U6I*V6I*V7I*U8*V10*U10*V10I*U10I*V9*U9*V8*V7*V6*V5*V4*V3*V2*V1;
Gc8I - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*U9I*V9I*U10*V10*U10I*V10I*U8I*V7*V6*U6*V6I*U6I*V5*U5*V4*V3*V2*V1;

Gc9 - V1I*V2I*V3I*V4I*V5I*V6I*U7I*V7I*U8*V10*U10*V10I*U10I*V9*U9*V8*V7*V6*V5*V4*V3*V2*V1;
Gc9I - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*U9I*V9I*U10*V10*U10I*V10I*U8I*V7*U7*V6*V5*V4*V3*V2*V1;

Gc10 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*V10*V9*V8*V7*V6*V5*V4*V3*V2*V1;
Gc10I - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*V10I*V9*V8*V7*V6*V5*V4*V3*V2*V1;

Gc11 - V1I*V2I*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I*U8I*V7*U7*V7I*U7I*V6*U6*V6I*U6I*V5*U5*V5I*V4*V3*V2*V1;
Gc11I - V1I*V2I*V3I*V4I*V5*U5I*V5I*U6*V6*U6I*V6I*U7*V7*U7I*V7I*U8*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V2*V1;

Gc12 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9*V8*V7*V6*V5*V4*V3*V2*V1;
Gc12I - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*V8*V7*V6*V5*V4*V3*V2*V1;

Gc13 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*U10I*V9*V8*V7*V6*V5*V4*V3*V2*V1;
Gc13I - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*U10*V9*V8*V7*V6*V5*V4*V3*V2*V1;

Gc14 - V1I*V2I*V3I*U4I*V4I*V5I*U6*V5*V4*V3*V2*V1;
Gc14I - V1I*V2I*V3I*V4I*V5I*U6I*V5*V4*U4*V3*V2*V1;

Gc15 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V7*U7I*V7I*U8*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V3*U3I*V3I*U4*V4*U4I
        *V4I*U5*V5*U5I*V5I*U6*V6*U6I*V6I*U7*V7*U7I*V7I*U8*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V1;
Gc15I - V1I*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I*U8I*V7*U7*V7I*U7I*V6*U6*V6I*U6I*V5*U5*V5I*U5I*V4*U4*V4I*U4I*V3*U3*V3I*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I
         *U8I*V7*U7*V7I*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I*V9*V8*V7*V6*V5*V4*V3*V2*V1;

Gc16 - U1I*V1I*V2I*V3I*U4*V4*U4I*V4I*V5I*U6*V6*V5*V4*V3*V2*V1;
Gc16I - V1I*V2I*V3I*V4I*V5I*V6I*U6I*V5*V4*U4*V4I*U4I*V3*V2*V1*U1;

Gc17 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V9I*U10*V9*V8*V7*V6*V5*V4*V3*V2*V1;
Gc17I - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*U10I*V9*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V7*V6*V5*V4*V3*V2*V1;

Gc18 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*U10I*V9*U9*V9*U9I*V9I*U10*V10*U10I*V10I*U8I*V7*V6*U6*V6I*U6I*V5*V4*U4*V4I*U4I*V3*U3*V2*V1;
Gc18I - V1I*V2I*U3I*V3I*U4*V4*U4I*V4I*V5I*U6*V6*U6I*V6I*V7I*U8*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V9I*U10*V9*V8*V7*V6*V5*V4*V3*V2*V1;

Gc19 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*U9I*V9I*U10*V10*V9*V8*V7*V6*V5*V4*V3*V2*V1;
Gc19I - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*V10I*U10I*V9*U9*V8*V7*V6*V5*V4*V3*V2*V1;

Gc20 - V1I*V2I*V3I*V4I*V5I*V6I*V7I*V8I*V9I*U10I*V10I*V8*U8I*V8I*U9*V9*U9I*V9I*U10*V10*U10I*V10I*V4*V3*V2*V1;
Gc20I - V1I*V2I*V3I*V4I*V10*U10*V10I*U10I*V9*U9*V9I*U9I*V8*U8*V8I*V10*U10*V9*V8*V7*V6*V5*V4*V3*V2*V1; 

// End of calculations for Proposition 4.
// **************************************

// Calculations for Proposition 5.
// *******************************
// For Proposition 5(a), the following routine quickly find $\beta$, $\nu$ and $\pi$ so
// that $\pi(b^\mu(M_\alpha))=b^\nu(M_\beta)$, for all $\mu\in{0,1,-1\}
// and $\alpha$ of the form $c_{\pm\pm\pm}$ EXCEPT that for $(\mu,\alpha)=
// $(1,c_{++-})$, and $(\mu,\alpha)=(-1,c_{-+-})$ for which the elements
// $\pi$ found have length 7 and 9, respectively. The routine after this one is
// much slower, but finds $\nu$, $\beta$ and $\pi$ in each case.

genlist:=[MatsL | ID3,A1,A1I,A2,A2I,A3,A3I];
genletters:=[[],["A1"],["A1I"],["A2"],["A2I"],["A3"],["A3I"]];
possiblefollowingletters:=[
[1,2,3,4,5,6,7],
[2,4,5,6,7],
[3,4,5,6,7],
[2,3,4,6,7],
[2,3,5,6,7],
[2,3,4,5,6],
[2,3,4,5,7]];

c:=(r-1)*(z^3-1)/2;

nubetalist:=[[0,4],[0,8],[1,4],[-1,4]];

kalphalist:=[MatsL | kppp,kppm,kpmp,kpmm,kmpp,kmpm,kmmp,kmmm];
alphalist:=["ppp","ppm","pmp","pmm","mpp","mpm","mmp","mmm"];

// for mu in [0,1,-1] do
//  for alpha in [1 .. 8] do
//   nubetafound:=0;
//   for idx in [1 .. 4] do
//    nu:=nubetalist[idx][1];
//    beta:=nubetalist[idx][2];
//     firstlist:=[MatsL | kalphalist[beta]^-1*B^(-nu)*ID3,kalphalist[beta]^-1*B^(-nu)*A1,kalphalist[beta]^-1*B^(-nu)*A1I,kalphalist[beta]^-1*B^(-nu)*A2,kalphalist[beta]^-1*B^(-nu)*A2I,kalphalist[beta]^-1*B^(-nu)*A3,kalphalist[beta]^-1*B^(-nu)*A3I];
//     lastlist:=[MatsL | B^mu*kalphalist[alpha],A1*B^mu*kalphalist[alpha],A1I*B^mu*kalphalist[alpha],A2*B^mu*kalphalist[alpha],A2I*B^mu*kalphalist[alpha],A3*B^mu*kalphalist[alpha],A3I*B^mu*kalphalist[alpha]];
//     for i1 in [1 .. 7] do
//      for i2 in possiblefollowingletters[i1] do
//       tmpmat1:=firstlist[i1]*genlist[i2];
//       for i3 in possiblefollowingletters[i2] do
//        tmpmat2:=tmpmat1*genlist[i3];
//        for i4 in possiblefollowingletters[i3] do
//         tmpmat3:=tmpmat2*genlist[i4];
//         for i5 in possiblefollowingletters[i4] do
//          tmpmat4:=tmpmat3*genlist[i5];
//          for i6 in possiblefollowingletters[i5] do
//           MM:=tmpmat4*lastlist[i6];
//           if ((MM[1][3] eq c*MM[2][3]) and (MM[1][2] eq c*(c*MM[2][1]-MM[1][1]+MM[2][2]))) then // Test whether $m$ is in $\bar\Gamma_c$ for $m=k_\beta^-1\pi b^\mu k_\alpha$
//            print "For mu=",mu," and alpha=",alphalist[alpha];
//            print "The solution is nu=",nu,"and beta=",alphalist[beta]," and pi equal to";
//            print genletters[i1] cat genletters[i2] cat genletters[i3] cat genletters[i4] cat genletters[i5] cat genletters[i6];
//            nubetafound:=1;
//            break idx;
//           end if;
//          end for;
//         end for;
//        end for;
//       end for;
//      end for;
//     end for;
//    end for;
//    if (nubetafound eq 0) then print "For mu=",mu,"and alpha=",alphalist[alpha]," we need a longer word"; end if;
//   end for;  
//  end for;

// Here is the same routine, except that we look for words of length at most 9 in the generators of $\Pi$:

// for mu in [0,1,-1] do
//  for alpha in [1 .. 6] do
//   nubetafound:=0;
//   for idx in [1 .. 4] do
//    nu:=nubetalist[idx][1];
//    beta:=nubetalist[idx][2];
//    firstlist:=[MatsL | kalphalist[beta]^-1*B^(-nu)*ID3,kalphalist[beta]^-1*B^(-nu)*A1,kalphalist[beta]^-1*B^(-nu)*A1I,kalphalist[beta]^-1*B^(-nu)*A2,kalphalist[beta]^-1*B^(-nu)*A2I,kalphalist[beta]^-1*B^(-nu)*A3,kalphalist[beta]^-1*B^(-nu)*A3I];
//    lastlist:=[MatsL | B^mu*kalphalist[alpha],A1*B^mu*kalphalist[alpha],A1I*B^mu*kalphalist[alpha],A2*B^mu*kalphalist[alpha],A2I*B^mu*kalphalist[alpha],A3*B^mu*kalphalist[alpha],A3I*B^mu*kalphalist[alpha]];
//    for i1 in [1 .. 7] do
//     for i2 in possiblefollowingletters[i1] do
//      tmpmat1:=firstlist[i1]*genlist[i2];
//      for i3 in possiblefollowingletters[i2] do
//       tmpmat2:=tmpmat1*genlist[i3];
//       for i4 in possiblefollowingletters[i3] do
//        tmpmat3:=tmpmat2*genlist[i4];
//        for i5 in possiblefollowingletters[i4] do
//         tmpmat4:=tmpmat3*genlist[i5];
//         for i6 in possiblefollowingletters[i5] do
//          tmpmat5:=tmpmat4*genlist[i6];
//          for i7 in possiblefollowingletters[i6] do
//           tmpmat6:=tmpmat5*genlist[i7];
//           for i8 in possiblefollowingletters[i7] do
//            tmpmat7:=tmpmat6*genlist[i8];
//            for i9 in possiblefollowingletters[i8] do
//             MM:=tmpmat7*lastlist[i9];
//             if ((MM[1][3] eq c*MM[2][3]) and (MM[1][2] eq c*(c*MM[2][1]-MM[1][1]+MM[2][2]))) then // Test whether $m$ is in $\bar\Gamma_c$ for $m=k_\beta^-1\pi b^\mu k_\alpha$
//              print "For mu=",mu," and alpha=",alphalist[alpha];
//              print "The solution is nu=",nu,", beta=",alphalist[beta]," and pi equal to";
//              print genletters[i1] cat genletters[i2] cat genletters[i3] cat genletters[i4] cat genletters[i5] cat genletters[i6] cat genletters[i7] cat genletters[i8] cat genletters[i9];
//              nubetafound:=1;
//              break idx;
//             end if;
//            end for;
//           end for;
//          end for;
//         end for;
//        end for;
//       end for;
//      end for;
//     end for;
//    end for;
//   end for;
//   if (nubetafound eq 0) then print "For mu=",mu,"and alpha=",alphalist[alpha]," we need a longer word"; end if;
//  end for;  
// end for;

// // Output of the quick routine:
// // ***************************
// For mu= 0  and alpha= ppp
// The solution is nu= 0 and beta= mmm  and pi equal to
// [ A2, A2 ]
// For mu= 0  and alpha= ppm
// The solution is nu= 0 and beta= pmm  and pi equal to
// [ A1, A3I ]
// For mu= 0  and alpha= pmp
// The solution is nu= 0 and beta= mmm  and pi equal to
// [ A3I, A1, A2, A2 ]
// For mu= 0  and alpha= pmm
// The solution is nu= 0 and beta= pmm  and pi equal to
// []
// For mu= 0  and alpha= mpp
// The solution is nu= 0 and beta= pmm  and pi equal to
// [ A1, A3I, A1, A3 ]
// For mu= 0  and alpha= mpm
// The solution is nu= 0 and beta= mmm  and pi equal to
// [ A2, A2, A1I, A3I ]
// For mu= 0  and alpha= mmp
// The solution is nu= 0 and beta= pmm  and pi equal to
// [ A1I, A2I, A2I, A1I ]
// For mu= 0  and alpha= mmm
// The solution is nu= 0 and beta= mmm  and pi equal to
// []
// For mu= 1  and alpha= ppp
// The solution is nu= 0 and beta= pmm  and pi equal to
// [ A3I, A3I, A3I ]
// For mu= 1 and alpha= ppm  we need a longer word
// For mu= 1  and alpha= pmp
// The solution is nu= 0 and beta= mmm  and pi equal to
// [ A2, A1I, A2I, A2I, A2I ]
// For mu= 1  and alpha= pmm
// The solution is nu= 1 and beta= pmm  and pi equal to
// []
// For mu= 1  and alpha= mpp
// The solution is nu= 0 and beta= pmm  and pi equal to
// [ A1I, A2I, A1, A3, A2I ]
// For mu= 1  and alpha= mpm
// The solution is nu= -1 and beta= pmm  and pi equal to
// [ A3, A1, A2I ]
// For mu= 1  and alpha= mmp
// The solution is nu= 0 and beta= mmm  and pi equal to
// [ A2, A1I, A2I ]
// For mu= 1  and alpha= mmm
// The solution is nu= 0 and beta= pmm  and pi equal to
// [ A1I, A2I, A2I ]
// For mu= -1  and alpha= ppp
// The solution is nu= 0 and beta= mmm  and pi equal to
// [ A2, A1I, A1I, A3I, A3I, A3I ]
// For mu= -1  and alpha= ppm
// The solution is nu= 0 and beta= mmm  and pi equal to
// [ A1I, A1I, A3I, A3I, A3I, A2I ]
// For mu= -1  and alpha= pmp
// The solution is nu= -1 and beta= pmm  and pi equal to
// [ A2, A2, A1, A3 ]
// For mu= -1  and alpha= pmm
// The solution is nu= -1 and beta= pmm  and pi equal to
// []
// For mu= -1  and alpha= mpp
// The solution is nu= 1 and beta= pmm  and pi equal to
// [ A3, A3, A3, A1, A2I ]
// For mu= -1 and alpha= mpm  we need a longer word
// For mu= -1  and alpha= mmp
// The solution is nu= 0 and beta= pmm  and pi equal to
// [ A1I, A2I, A2I, A2I ]
// For mu= -1  and alpha= mmm
// The solution is nu= 0 and beta= pmm  and pi equal to
// [ A1I, A2I ]
// 
// // Output of the slower routine:
// // *****************************
// For mu= 0  and alpha= ppp
// The solution is nu= 0 , beta= mmm  and pi equal to
// [ A2, A2 ]
// For mu= 0  and alpha= ppm
// The solution is nu= 0 , beta= pmm  and pi equal to
// [ A1, A3I ]
// For mu= 0  and alpha= pmp
// The solution is nu= 0 , beta= mmm  and pi equal to
// [ A3I, A1, A2, A2 ]
// For mu= 0  and alpha= pmm
// The solution is nu= 0 , beta= pmm  and pi equal to
// []
// For mu= 0  and alpha= mpp
// The solution is nu= 0 , beta= pmm  and pi equal to
// [ A1, A3I, A1, A3 ]
// For mu= 0  and alpha= mpm
// The solution is nu= 0 , beta= mmm  and pi equal to
// [ A2, A2, A1I, A3I ]
// For mu= 1  and alpha= ppp
// The solution is nu= 0 , beta= pmm  and pi equal to
// [ A3I, A3I, A3I ]
// For mu= 1  and alpha= ppm
// The solution is nu= 0 , beta= mmm  and pi equal to
// [ A1I, A2I, A3I, A3I, A1, A1, A2I ]
// For mu= 1  and alpha= pmp
// The solution is nu= 0 , beta= mmm  and pi equal to
// [ A2, A1I, A2I, A2I, A2I ]
// For mu= 1  and alpha= pmm
// The solution is nu= 1 , beta= pmm  and pi equal to
// []
// For mu= 1  and alpha= mpp
// The solution is nu= 0 , beta= pmm  and pi equal to
// [ A1I, A2I, A1, A3, A2I ]
// For mu= 1  and alpha= mpm
// The solution is nu= -1 , beta= pmm  and pi equal to
// [ A3, A1, A2I ]
// For mu= -1  and alpha= ppp
// The solution is nu= 0 , beta= mmm  and pi equal to
// [ A2, A1I, A1I, A3I, A3I, A3I ]
// For mu= -1  and alpha= ppm
// The solution is nu= 0 , beta= mmm  and pi equal to
// [ A1I, A1I, A3I, A3I, A3I, A2I ]
// For mu= -1  and alpha= pmp
// The solution is nu= -1 , beta= pmm  and pi equal to
// [ A2, A2, A1, A3 ]
// For mu= -1  and alpha= pmm
// The solution is nu= -1 , beta= pmm  and pi equal to
// []
// For mu= -1  and alpha= mpp
// The solution is nu= 1 , beta= pmm  and pi equal to
// [ A3, A3, A3, A1, A2I ]
// For mu= -1  and alpha= mpm
// The solution is nu= 1 , beta= pmm  and pi equal to
// [ A2, A1I, A1I, A3I, A1, A3I, A1I, A2I, A2I ]

// End of calculations for Proposition 5:
// **************************************

// Calculations for the last bit of Section 2.4.
// *********************************************
// We give generators of $\Pi_M$ for $M=b(M_c)$. While we're at it, we'll 
// also give generators of $\Pi_M$ for $M=b^{-1}(M_c)$.

// Here are the elements $p_1,\ldots,p_8$ given in Section 2.4,
// though here we have multiplied them by scalars so that
// the matrices $g=b^{-1}p_jb$, which are in $\bar\Gamma_c$, as
// we see below, satisfy the normalization $g_{11}-c*g_{21}=1$.

P1:=(1/z)*A2*A2*A2*A1I*A3I*J^8*A2I*A2I*A1I*J^4;
P2:=A3*A3*A3*A1*A3*A3*A2*A1*J^4*A3I*J^8*A3I*A3I*A1I*A3I*A3I*A3I;
P3:=z^4*J^8*A1I*A3I*A3I*A3I*A2*A2*J^4*A3I*A3I*A1I*A3I*A3I*A3I;
P4:=(1/z^2)*J^8*A2*A1*A2I*A2I*A1I*J^4*A3*A3*A3*A1*A1*A2I;
P5:=(1/z^4)*A3*A3*A3*A1*A3*A3*J^4*A1I*J^8*A3*A3*A1*A2I*A2I*A2I;
P6:=z^4*A3*A3*A3*A1*A2*A1*A3*A2I*A2I*A2I;
P7:=(1/z)*A3*A3*A3*A1*J^8*A1*A2I*A2I*A1I*A3*A3*J^4;
P8:=(1/z^2)*J^4*A3I*A3I*J^8*A2*A1*A2*A1*A2I*A2I;

P1I:=P1^-1;
P2I:=P2^-1;
P3I:=P3^-1;
P4I:=P4^-1;
P5I:=P5^-1;
P6I:=P6^-1;
P7I:=P7^-1;
P8I:=P8^-1;

// The following is the identity matrix:
P5I*P2I*P5*P1*P3*P8I*P4*P1I*P7I*P6I*P7*P2*P3I*P8*P4I*P6;

Plist:=[MatsL | P1,P2,P3,P4,P5,P6,P7,P8];

// The following gives an output of 24 0's, checking that the $p_j$ 
// satisfy $g=b^{-1}p_jb\in\bar\Gamma_c$, and also the normalization $g_{11}-cg_{21}=1$.
for jj in [1 .. #Plist] do
 tmpmat:=BI*Plist[jj]*B; 
 print tmpmat[1][1]-c*tmpmat[2][1] - 1, tmpmat[1][3]-c*tmpmat[2][3], tmpmat[1][2]-c*(c*tmpmat[2][1]-tmpmat[1][1]+tmpmat[2][2]);
end for;

// We can also see that the $b^{-1}p_jb$'s are in $\bar\Gamma_c$ by
// exhibiting each of them as a word in the generators $t_2$, $t_4$, 
// $t_{12}$ and $z_c$ given in Section 2.1.

// The following are zero:
BI*P1*B - z^4*t12^9*t2*t12^11*t2*t12^8*zc;
BI*P2*B - z^4*t12^9*t2*t12^3*t2*t12^4*zc;
BI*P3*B - z^4*t12*t2*t12^11*t2*t12^4*zc;
BI*P4*B - z^8*t2*t12*t2*t12^7*zc^2;
BI*P5*B - t12^8*t2*t12^11*t2*t12^9*t2*t12^2;
BI*P6*B - z^4*t12^11*t2*t12^3*t2*t12^2*zc;
BI*P7*B - z^8*t12^10*t2*t12^11*t2*t12^2*t2*t12^3*zc^2;
BI*P8*B - z^8*t2*t12^11*t2*t12^10*t2*t12^5*zc^2;

// To see that $p_1,\ldots,p_8$ generate $\Pi_M$ for $M=b(M_c)$,
// we check that the subgroup of $\bar\Gamma_c$ generated by
// the $b^{-1}p_jb$'s generates a subgroup of $\bar\Gamma_c$
// of index 108, containing $\Pi_c$ as a subgroup of index 3,
// in accordance with Proposition 5(d). We'll do that in the
// other magma file.

// Changing the generators of $\Pi_M$, where $M=b(M_c)$, from $p_1,\ldots,p_8$
// to variables $u_i$ and $v_i$, $i=1,\ldots,4$, which satisfy the standard 
// surface group relation:
// Set
D1:=P5I*P2I*P5*P1*P3*P8I*P4*P1I*P7I;
D2:=P5I*P2I*P5*P1*P3*P8I;
D3:=P5I*P2I*P5*P1;
D4:=P5I;

E1:=P6I; 
E2:=P4*P1I*P2*P3I; 
E3:=P3;          
E4:=P2I;

U1:=D1;                            V1:=E1;
U2:=E1*D2*E1^-1;                   V2:=E1*E2*E1^-1;
U3:=E1*E2*D3*E2^-1*E1^-1;          V3:=E1*E2*E3*E2^-1*E1^-1;
U4:=E1*E2*E3*D4*E3^-1*E2^-1*E1^-1; V4:=E1*E2*E3*E4*E3^-1*E2^-1*E1^-1;

U1I:=U1^-1;
U2I:=U2^-1;
U3I:=U3^-1;
U4I:=U4^-1;

V1I:=V1^-1;
V2I:=V2^-1;
V3I:=V3^-1;
V4I:=V4^-1;

ID3 - (U1*V1*U1^-1*V1^-1)*(U2*V2*U2^-1*V2^-1)*(U3*V3*U3^-1*V3^-1)*(U4*V4*U4^-1*V4^-1);   // ID3 is a product of four commutators

print "ID3 is a product of four commutators";

// Expressing the old generators Pi and their inverses in terms of the new generators Ui and Vi:
// The following are zero:
P1 - V1I*V2I*V3I*U4*V4I*U4I*V3*U3*V2*V1;
P1I - V1I*V2I*U3I*V3I*U4*V4*U4I*V3*V2*V1;

P2 - V1I*V2I*V3I*V4I*V3*V2*V1;
P2I - V1I*V2I*V3I*V4*V3*V2*V1;

P3 - V1I*V2I*V3*V2*V1;
P3I - V1I*V2I*V3I*V2*V1;

P4 - V1I*V4*U4*V4I*U4I*V3*U3*V2*V1;
P4I - V1I*V2I*U3I*V3I*U4*V4*U4I*V4I*V1;

P5 - V1I*V2I*V3I*U4I*V3*V2*V1;
P5I - V1I*V2I*V3I*U4*V3*V2*V1;

P6 - V1I;
P6I - V1;

P7 - U1I*V1I*U2*V4*V3*V2*V1;
P7I - V1I*V2I*V3I*V4I*U2I*V1*U1;

P8 - V1I*U2I*V2I*U3*V3*V2*V1;
P8I - V1I*V2I*V3I*U3I*V2*U2*V1;

// Here are the generators of $\Pi_M$ for $M=b^{-1}(M_c)$:
M1:=z^2*J^8*A1*A3I*A1I*A2I*A2I*J^8*A2*A1*J^8;
M2:=(1/z^3)*J^8*A1*A3*A3*A3*J^4;
M3:=(1/z^5)*J^4*A2I*A2I*A1I*A3I*J^8*A1*A3I*A3I*A1I*A2I*A2I;
M4:=(-1)*J^8*A1*A3I*A1I*A2*A1I*A2I*J^4*A2I*A2I*A3*A3*A3*A1;
M5:=z^4*J^4*A1I*A1I*A3I*A3I*A3I*J^4*A1I*A2I*J^4*A2I*A3*A3*A3*A1;
M6:=(1/z^4)*J^4*A1I*A2I*A1*A3*A3*A3*A1*A2I*J^4*A1I*A2I*A2I*A3*A3*A3*J^4;
M7:=z^2*A3*A1*A2*J^4*A2*A1*J^8;
M8:=(-1)*J^4*A1I*A2I*A1*A2*A2*A1*J^8*A2*A3I*A1I*A2I*A2I;

M1I:=M1^-1;
M2I:=M2^-1;
M3I:=M3^-1;
M4I:=M4^-1;
M5I:=M5^-1;
M6I:=M6^-1;
M7I:=M7^-1;
M8I:=M8^-1;

// The following is the identity matrix:
M3*M8I*M4*M5*M7I*M2*M3I*M1*M5I*M7*M4I*M1I*M6*M2I*M6I*M8;

Mlist:=[MatsL | M1,M2,M3,M4,M5,M6,M7,M8];
// The following gives an output of 24 0's, checking that the $m_j$'s 
// satisfy $g=bm_jb^{-1}\in\bar\Gamma_c$, and also the normalization $g_{11}-cg_{21}=1$.
for jj in [1 .. #Mlist] do
 tmpmat:=B*Mlist[jj]*BI; 
 print tmpmat[1][1]-c*tmpmat[2][1] - 1, tmpmat[1][3]-c*tmpmat[2][3], tmpmat[1][2]-c*(c*tmpmat[2][1]-tmpmat[1][1]+tmpmat[2][2]);
end for;

// We can also see that the $bm_jb^{-1}$'s are in $\bar\Gamma_c$ by
// exhibiting each of them as a word in the generators $t_2$, $t_4$, 
// $t_{12}$ and $z_c$ given in Section 2.1.
// The following are zero:
B*M1*BI - z^4*t12^8*t2*t12^2*zc;
B*M2*BI - z^4*t2*t12^10*zc;
B*M3*BI - z^8*t12^6*t2*t12^8*zc^2;
B*M4*BI - t12^9*t2*t12^10*t2*t12^10*t2*t12;
B*M5*BI - t12^11*t2*t12^7*t2*t12^11*t2*t12;
B*M6*BI - t12^9*t2*t12^4*t2*t12^11;
B*M7*BI - z^4*t2*t12*t2*t12^3*t2*t12^11*t2*t12*zc;
B*M8*BI - t12^9*t2*t12^8*t2*t12^7;

// To see that $m_1,\ldots,m_8$ generate $\Pi_M$ for $M=b^{-1}(M_c)$,
// we check that the subgroup of $\bar\Gamma_c$ generated by
// the $bm_jb^{-1}$'s generates a subgroup of $\bar\Gamma_c$
// of index 108, containing $\Pi_c$ as a subgroup of index 3,
// in accordance with Proposition 5(d). We'll do that in the
// other magma file.

// Changing the generators of $\Pi_M$, where $M=b^{-1}(M_c)$, from $m_1,\ldots,m_8$
// to variables $u_i$ and $v_i$, $i=1,\ldots,4$, which satisfy the standard 
// surface group relation:

D1:=M3;
D2:=M1*M5I*M7*M4I*M1I*M6;
D3:=M1*M5I;
D4:=M1;

E1:=M8I*M4*M5*M7I*M2;
E2:=M2I;
E3:=M7*M4I*M1I*M4;
E4:=M4I;

// Then we define $u_i$ and $v_i$ by setting:
U1:=D1;
V1:=E1;

U2:=E1*D2*E1^-1;
V2:=E1*E2*E1^-1;

U3:=E1*E2*D3*E2^-1*E1^-1;
V3:=E1*E2*E3*E2^-1*E1^-1;

U4:=E1*E2*E3*D4*E3^-1*E2^-1*E1^-1;
V4:=E1*E2*E3*E4*E3^-1*E2^-1*E1^-1;

U1I:=U1^-1;
U2I:=U2^-1;
U3I:=U3^-1;
U4I:=U4^-1;

V1I:=V1^-1;
V2I:=V2^-1;
V3I:=V3^-1;
V4I:=V4^-1;

// The following is zero:
ID3 - (U1*V1*U1I*V1I)*(U2*V2*U2I*V2I)*(U3*V3*U3I*V3I)*(U4*V4*U4I*V4I);   // ID3 is a product of four commutators

print "ID3 is a product of four commutators";

// Expressing the original generators $M_i$ in terms of the new
// generators $u_i$ and $v_i$. The following are zero:

M1 - V1I*V2I*V3I*U4*V3*V2*V1;
M1I - V1I*V2I*V3I*U4I*V3*V2*V1;

M2 - V1I*V2I*V1;
M2I - V1I*V2*V1; 

M3 - U1;
M3I - U1I;

M4 - V1I*V2I*V3I*V4I*V3*V2*V1;
M4I - V1I*V2I*V3I*V4*V3*V2*V1;

M5 - V1I*V2I*U3I*V3I*U4*V3*V2*V1;
M5I - V1I*V2I*V3I*U4I*V3*U3*V2*V1;

M6 - V1I*V2I*V3I*V4I*U3I*V2*U2*V1;
M6I - V1I*U2I*V2I*U3*V4*V3*V2*V1;

M7 - V1I*V2I*V4*U4*V4I*V3*V2*V1;
M7I - V1I*V2I*V3I*V4*U4I*V4I*V2*V1;

M8 - V1I*V2I*V3I*V4I*V3*U3I*V3I*U4*V4*U4I*V4I;
M8I - V4*U4*V4I*U4I*V3*U3*V3I*V4*V3*V2*V1;

// End of calculations needed for the last bit of Section 2.4.
// ***********************************************************

// Calculations for Proposition 6:
// ******************************
// It is easy to see that $n_\mu(\pi(M))=n_\mu(M)$ for any mirror $M$ and any $\pi\in\Pi$.
// so we need only consider the cases $M=M_0$, $M=M_1$ and $M=M_\infty$.
// It is also easy to see that the number $n_\mu(M_\alpha)$ of $\Pi_\alpha \xi\in \Pi_\alpha\backslash M_\alpha$
// such that $\Pi\xi=\Pi(b^\mu O)$ is equal to the number of double cosets $\Pi_\alpha g K_\alpha$,
// where $g\in\bar\Gamma_\alpha$, such that $g\in\Pi b^\mu K$.
// In this proposition, $\alpha\in\{0,1,\infty\}$, and we first find representatives $g\in\bar\Gamma_\alpha$ 
// for the six double cosets $\Pi_\alpha g K_\alpha$.

// The case $\alpha=0$. 
// ********************
// We have already seen in the calculations for Proposition 3(c) that the following six elements
rep01:=ID3;
rep02:=J^9*B*J^7;
rep03:=J^10*B*J^7;
rep04:=J^11*B*J^7;
rep05:=B*J^7;
rep06:=J^9*B*J^2*BI*J;
// are representatives for the six double cosets $\Pi_0 g K_0$. 
// We can see which of these representatives $g$ lie in $\Pi b^\mu K$
// using Magma's Index routine. Alternatively, we can see this by
// noting that the following are zero:
rep01 - B^0;
rep02 - z^4*(J^4*A3I*A3I*A3I*A1I*A2*A1*J^8)*B^(-1)*(V*J^2);
rep03 - (1/z)*(A1I*A2I)*B^0*(V*U*J^8);
rep04 - (1/z^4)*(A1I*A1I*A3I*A3I*A3I)*B^(-1)*(UI*V^2*U*J);
rep05 - B^1*J^7;
rep06 - (1/z^4)*(J^4*A3I*A3I*A3I*J^8)*B^0*(U*V^2*J^4);
// So $n_0(M_0)=3$, $n_1(M_0)=1$ and $n_{-1}(M_0)=2$.

// The cases $\alpha=1,\infty$. 
// ***************************
// Since $M_\alpha=k_\alpha(M_0)$ for $\alpha=1,\infty$, we get
// $\bar\Gamma_\alpha=k_\alpha\bar\Gamma_0k_\alpha^{-1}$, 
// $\Pi_\alpha=k_\alpha\Pi_0k_\alpha^{-1}$ and $K_\alpha=k_\alpha K_0k_\alpha^{-1}$.
// and the six elements $rep_{\alpha i}=k_\alpha rep_{0i}k_\alpha^{-1}$
// are representatives for the double cosets $\Pi_\alpha g K_\alpha$.

kplus1:=V^3*U*J;

rep11:=kplus1*rep01*kplus1^-1;
rep12:=kplus1*rep02*kplus1^-1;
rep13:=kplus1*rep03*kplus1^-1;
rep14:=kplus1*rep04*kplus1^-1;
rep15:=kplus1*rep05*kplus1^-1;
rep16:=kplus1*rep06*kplus1^-1;

// The following are zero:
rep11 - B^0;
rep12 - (1/z)*(A2I*A2I)*B^1*(U*VI*J^2);
rep13 - (1/z)*(A2I)*B^1*(VI*U*VI*J^3);
rep14 - z^3*(A3*A3*A3*A1*A1*A2I)*B^1*(U*V*V*UI);
rep15 - (1/z^4)*(A1I*A3I*A3I*A3I)*B^1*(V*J^4);
rep16 - z^2*(A2I*A2I*A3*A3*A3*A1*A3I*A1I*A2I*A2I)*B^(-1)*(V*U*VI*J^3);
// So $n_0(M_1)=1$, $n_1(M_1)=4$ and $n_{-1}(M_1)=1$.

kinfty:=UI*V^2*U*J^6;
repinfty1:=kinfty*rep01*kinfty^-1;
repinfty2:=kinfty*rep02*kinfty^-1;
repinfty3:=kinfty*rep03*kinfty^-1;
repinfty4:=kinfty*rep04*kinfty^-1;
repinfty5:=kinfty*rep05*kinfty^-1;
repinfty6:=kinfty*rep06*kinfty^-1;

// The following are zero:
repinfty1 - B^0;
repinfty2 - (1/z)*(J^8*A1I*J^4)*B^(-1)*(V^2*UI*J^10);
repinfty3 - (A2I*A2I)*B^(-1)*(U*V^2*J^7);
repinfty4 - (1/z)*(A2I)*B^0*(U*V^2*J^6);
repinfty5 - (-1)*(J^8*A3*A1I*A3I*A3I*A3I*A1I*J^4)*B^1*(V^2*J^4);
repinfty6 - (1/z^2)*(J^8*A1I*A3I*J^4)*B^(-1)*(V*U*J^6);
// So $n_0(M_\infty)=2$, $n_1(M_\infty)=1$ and $n_{-1}(M_\infty)=3$.

// End of calculations for Proposition 6.
// **************************************

// Calculations for Proposition 7:
// *******************************
// As noted for Proposition 6, $n_\mu(\pi(M))=n_\mu(M)$ for any mirror $M$ and any $\pi\in\Pi$.
// so we need only consider the cases $M=M_c$, $M=M_{-c}$, $M=b(M_c)$ and $M=b^{-1}(M_c)$.

// Also, as for Proposition 6, the number $n_\mu(M_\alpha)$ of $\Pi_\alpha \xi\in \Pi_\alpha\backslash M_\alpha$
// such that $\Pi\xi=\Pi(b^\mu O)$ is equal to the number of double cosets $\Pi_\alpha g K_\alpha$,
// where $g\in\bar\Gamma_\alpha$, such that $g\in\Pi b^\mu K$.
// In this proposition, $\alpha\in\{c,-c\}$, and we first find representatives $g\in\bar\Gamma_\alpha$ 
// for the nine double cosets $\Pi_\alpha g K_\alpha$.
// Here are representatives for the double cosets $\Pi_c g K_c$, $g\in\bar\Gamma_c$:
repc1:=ID3;
repc2:=J^7*B*UI*B*U*J^7;
repc3:=J^8*B*UI*B*U*J^7;
repc4:=J^9*B*UI*B*U*J^7;
repc5:=J^10*B*UI*B*U*J^7;
repc6:=J^7*B*UI*B*U*J^11*B*UI*B*UI*J^2;
repc7:=J^8*B*UI*B*U*J^11*B*UI*B*UI*J^2;
repc8:=J^9*B*UI*B*U*J^11*B*UI*B*UI*J^2;
repc9:=J^10*B*UI*B*U*J^11*B*UI*B*UI*J^2;

// We now exhibit these representatives in the form $\pi b^\mu k$ for $\pi\in\Pi$,
// $\mu\in\{0,1,-1\}$ and $k\in K$.
// The following are zero: 
repc1 - B^0;
repc2 - z^3*(J^4*A2*A1*J^8*A3*A3*A3*A1*A1*A2I)*B^1*(V*UI*V*J^9);
repc3 - z^4*(J^4*A1*A3*A3*J^4*A1I*J^4)*B^1*(VI*U*VI);
repc4 - z*(A1I*A2I*A2I*A2I)*B^(-1)*(V*UI*J^9);
repc5 - (1/z^2)*(A1I*A2I)*B^(-1)*(V*V*U*J^6);
repc6 - (1/z^4)*(J^4*A1*A3*A3*A3*A1*A3I*J^8)*B^0*(V*UI*V*J^2);
repc7 - z^2*(J^4*A1*A3I*J^8)*B^0*(U*VI*J^5);
repc8 - z*(A1I*A2I*A2I)*B^1*(V*V*UI*J^2);
repc9 - z^3*(J^8*A2*A1*J^4*A3)*B^0*(V*U*J^7);
// So $n_0(M_c)=4$, $n_1(M_c)=3$ and $n_{-1}(M_c)=2$.

// Here are representatives for the double cosets $\Pi_{-c} g K_{-c}$, $g\in\bar\Gamma_{-c}$.
repmc1:=kmmm*repc1*kmmm^-1;
repmc2:=kmmm*repc2*kmmm^-1;
repmc3:=kmmm*repc3*kmmm^-1;
repmc4:=kmmm*repc4*kmmm^-1;
repmc5:=kmmm*repc5*kmmm^-1;
repmc6:=kmmm*repc6*kmmm^-1;
repmc7:=kmmm*repc7*kmmm^-1;
repmc8:=kmmm*repc8*kmmm^-1;
repmc9:=kmmm*repc9*kmmm^-1;
// We now exhibit these representatives in the form $\pi b^\mu k$ for $\pi\in\Pi$,
// $\mu\in\{0,1,-1\}$ and $k\in K$.
// The following are zero:
repmc1 - B^0;
repmc2 - (J^4*A1*A2*A2*J^8)*B^(-1)*(U*V*J^6);
repmc3 - (1/z^2)*(J^4*A3I*A3I*J^8)*B^0*(UI*V*V*U*J^7);
repmc4 - (A1I*A2I*A3I*A3I*A3I*J^8*A2*A1*A2I*A2I*A1I*J^4)*B^1*(V);
repmc5 - (1/z^4)*(J^8*A1I*A3I*A3I*A3I*A2*A2*J^4)*B^0*(U*VI*U*J^8);
repmc6 - z*(J^4*A3I*A2*A1*J^8*A2I)*B^(-1)*(UI*V*V*U*J^5);
repmc7 - (1/z)*(J^8*A2*A1*A3*A1I*J^4*A1I*A3I*A3I*A3I)*B^1*(UI*V*J);
repmc8 - (J^8*A1I*A3I*A3I*A3I*A2*A1I*J^4)*B^1*(V*UI*V*V*U);
repmc9 - z^4*(A2*J^4*A2*A1*A2I*A2I*J^8)*B^0*(V*J^6);
// So $n_0(M_{-c})=4$, $n_1(M_{-c})=3$ and $n_{-1}(M_{-c})=2$.

// We find (in the other Magma file) that $1$, $bt_2b^{-1}$ and 
// $bt_4b^{-1}$ form a set of representatives $\gamma\in\bar\Gamma_{b(M_c)}$
// for the distinct double cosets $\Pi_{b(M_c)}\gamma(bK_cb^{-1})$.
// We also find in the other Magma file the $\mu\in\{0,1,-1\}$
// and the $k\in K$ such that $\gamma b\in\Pi b^\mu k$, for each
// of these three $\gamma$'s. This is confirmed here by checking that
// the following are zero:
B*ID3 - B^1;
B*t2 - z^4*(J^8*A1*A3*A3*J^4)*B^(-1)*(VI*U*V^2*J^-1);
B*t4 - z^2*(A3*A3*A3*A1*A2I)*B^(-1)*(V*U*VI*J);
// So $n_0(M)=0$, $n_1(M)=1$ and $n_{-1}(M)=2$ for $M=b(M_c)$.

// We find (in the other Magma file) that $1$, $b^{-1}t_4b$ and 
// $b^{-1}t_2t_4b$ form a set of representatives $\gamma\in\bar\Gamma_{b^{-1}(M_c)}$
// for the distinct double cosets $\Pi_{b^{-1}(M_c)}\gamma(b^{-1}K_cb)$.
// We also find in the other Magma file the $\mu\in\{0,1,-1\}$
// and the $k\in K$ such that $\gamma b^{-1}\in\Pi b^\mu k$, for each
// of these three $\gamma$'s. This is confirmed here by noting that
// the following are zero:
B^(-1)*ID3 - B^(-1);
B^-1*t4 - z*(J^4*A1I*A1I*A3I*A3I*A3I*J^4*A1I*A2I*J^4)*B^1*(V*V*U*VI*J^5);
B^-1*t2*t4 - z^2*(J^8*A1*A2*A2*A1*A3*A1I*J^4)*B^(-1)*(VI*J^8);
// So $n_0(M)=0$, $n_1(M)=1$ and $n_{-1}(M)=2$ for $M=b^{-1}(M_c)$.

t2:=(B*U^-1)^2;
t4:=J^-1*t2;
t12:=J;
zc:=U;
// The following is a set of 54 representatives for the double cosets 
// $\Pi_c\gamma(\bar\Gamma_c\cap\bar\Gamma_P)$ in
// $\Pi_c\backslash\bar\Gamma_c/(\bar\Gamma_c\cap\bar\Gamma_P)$:
replist:=[MatsL | 
ID3,t4,t12,t4^-1,t2*t4,t4^2,t4*t12,t12^2,t12*t4^-1,t4^-1*t12,t12^-1*t4,t2*t4*t12,t2*t12^-1*t4,t4^2*t12,
t4*t12^2,t4*t12*t4^-1,t4*t12^-1*t4,t12*t2*t4,t12^3,t12^2*t4^-1,t4^-1*t2*t4,t4^-1*t12^2,t12^-1*t4*t12,t12^-2*t4,
t2*t4*t12^2,t2*t12^-1*t4*t12,t2*t12^-2*t4,t4^2*t12^2,t4*t12*t2*t4,t4*t12^3,(t4*t12*t4^-1)^2,t4*t12^-1*t4*t12,
t4*t12^-2*t4,t12*t2*t4*t12,t12*t2*t12^-1*t4,t12^2*t2*t4,t12^4,t12^3*t4^-1,t4^-1*t2*t12^-1*t4,t4^-1*t12*t2*t4,
t4^-1*t12^2*t4^-1,t12^-1*t4*t12^2,t12^-2*t4*t12,t12^-3*t4,t2*t4*t12^3,t2*t4*t12^2*t4^-1,t2*t4*t12^-2*t4,
t2*t12^-1*t4*t12^2,t2*t12^-2*t4*t12,t4^2*t12^3,t4^2*t12^2*t4^-1,t4*t12^-1*t4*t12^2,t4*t12^-3*t4,t12^-1*t4*t12^2*t4^-1];

// The following is a list of representatives for the double cosets $\Pi\gamma\bar\Gamma_P$
// in $\Pi\backslash\bar\Gamma/\bar\Gamma_P$. This is the list given at the beginning of 
// Section 2. The order of this list is chosen to make the table in Proposition 7 as neat as possible.
PibarGammabarGammaPdoublecosetreplist:=[MatsL |
V,
V*J^4,
V*J^8,
V^2,
V^2*J^4,
V^2*J^8,
V*U*VI,
V*U*VI*J^4,
V*U*VI*J^8,
V*UI*V^2*U,
V*UI*V^2*U*J^4,
V*UI*V^2*U*J^8,
VI,
VI*J^4,
VI*J^8,
U*V^2,
U*V^2*J^4,
U*V^2*J^8,
J,
J^5,
J^9,
J^2,
J^6,
J^10,
ID3,
J^4,
J^8,
J^3,
J^7,
J^11,
U*V,
U*V*J^4,
U*V*J^8,
UI*VI,
UI*VI*J^4,
UI*VI*J^8];
// The following are zero
replist[1] - (ID3)*PibarGammabarGammaPdoublecosetreplist[25]*Pfixerlist[1];
replist[2] - (ID3)*PibarGammabarGammaPdoublecosetreplist[30]*Pfixerlist[24];
replist[3] - (ID3)*PibarGammabarGammaPdoublecosetreplist[19]*Pfixerlist[1];
replist[4] - z^4*(J^8*A1*A3*A3*A3*A1*A3I*J^4)*PibarGammabarGammaPdoublecosetreplist[8]*Pfixerlist[23];
replist[5] - z^3*(J^8*A1*J^4)*PibarGammabarGammaPdoublecosetreplist[10]*Pfixerlist[13];
replist[6] - (J^8*A2*A1*A2I*A2I*A2I*J^8*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[9]*Pfixerlist[8];
replist[7] - (1/z)*(J^8*A2*A1*J^4*A3)*PibarGammabarGammaPdoublecosetreplist[2]*Pfixerlist[22];
replist[8] - (ID3)*PibarGammabarGammaPdoublecosetreplist[22]*Pfixerlist[1];
replist[9] - z^2*(J^4*A1I*A2I*A1*A3*A1I*J^8)*PibarGammabarGammaPdoublecosetreplist[21]*Pfixerlist[4];
replist[10] - z^2*(J^8*A1*A3*A3*A3*A1*A2I*J^4)*PibarGammabarGammaPdoublecosetreplist[12]*Pfixerlist[10];
replist[11] - (ID3)*PibarGammabarGammaPdoublecosetreplist[24]*Pfixerlist[24];
replist[12] - z^3*(J^8*A1*A3I*J^8*A2*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[31]*Pfixerlist[19];
replist[13] - z^5*(J^8*A1*A3I*J^8*A2*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[11]*Pfixerlist[18];
replist[14] - z^2*(J^8*A1*A2*A2*A1I*A2I*J^4*A3)*PibarGammabarGammaPdoublecosetreplist[19]*Pfixerlist[17];
replist[15] - z*(J^4*A2I*A2I*A2I*A3*A3*J^4*A1I*J^4)*PibarGammabarGammaPdoublecosetreplist[1]*Pfixerlist[12];
replist[16] - (1/z)*(J^8*A2*A1*J^8*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[22]*Pfixerlist[4];
replist[17] - z^2*(J^8*A2*A1*A2I*A2I*A2I*J^8*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[3]*Pfixerlist[17];
replist[18] - z*(J^4*A1I*A2I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[5]*Pfixerlist[4];
replist[19] - (ID3)*PibarGammabarGammaPdoublecosetreplist[28]*Pfixerlist[1];
replist[20] - (1/z)*(J^4*A1I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[5]*Pfixerlist[16];
replist[21] - (J^8*A2*A1*A2I*A2I*A1I*A3I*J^4)*PibarGammabarGammaPdoublecosetreplist[24]*Pfixerlist[13];
replist[22] - (1/z^3)*(J^8*A1*A3*A3*A3*A1*A2I*J^8*A1I*A3I*A3I*A3I*A2*A2*J^8)*PibarGammabarGammaPdoublecosetreplist[11]*Pfixerlist[9];
replist[23] - (1/z)*(A1I*A2I)*PibarGammabarGammaPdoublecosetreplist[4]*Pfixerlist[16];
replist[24] - (ID3)*PibarGammabarGammaPdoublecosetreplist[21]*Pfixerlist[24];
replist[25] - z^3*(J^8*A1*J^4*A2*A1I)*PibarGammabarGammaPdoublecosetreplist[20]*Pfixerlist[11];
replist[26] - (1/z^3)*(J^8*A3I*A3I*A3I*A2*A2*A1I*A2I*J^4)*PibarGammabarGammaPdoublecosetreplist[24]*Pfixerlist[19];
replist[27] - (1/z)*(J^8*A3I*A3I*A3I*A2*A2*A1I*J^4)*PibarGammabarGammaPdoublecosetreplist[7]*Pfixerlist[3];
replist[28] - (1/z^3)*(J^8*A1*A3*A3*A3*A1*A3I*J^8*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[22]*Pfixerlist[7];
replist[29] - (1/z)*(J^8*A2*A1*J^4*A3)*PibarGammabarGammaPdoublecosetreplist[35]*Pfixerlist[24];
replist[30] - z*(J^4*A2I*A2I*A2I*A3*A3*J^4*A1I*J^4)*PibarGammabarGammaPdoublecosetreplist[7]*Pfixerlist[15];
replist[31] - (J^8*A1*A2*A2*A1I*A2I*J^8*A1I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[4]*Pfixerlist[21];
replist[32] - z^4*(J^8*A2*A1*J^8*A3*A3*A3*A1*A3I*J^8)*PibarGammabarGammaPdoublecosetreplist[6]*Pfixerlist[21];
replist[33] - z^2*(A3*A3*A3*A2*A1*A2I*A2I*A2I)*PibarGammabarGammaPdoublecosetreplist[1]*Pfixerlist[2];
replist[34] - (1/z)*(J^4*A1I*A2I*A2I*A1I*J^8)*PibarGammabarGammaPdoublecosetreplist[36]*Pfixerlist[23];
replist[35] - z*(J^4*A1I*A2I*A2I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[2]*Pfixerlist[12];
replist[36] - z*(J^8*A2I*A1*A3*A1I*A3I*J^8*A1I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[23]*Pfixerlist[5];
replist[37] - (ID3)*PibarGammabarGammaPdoublecosetreplist[26]*Pfixerlist[1];
replist[38] - (1/z^2)*(J^8*A2I*A1*A3*A1I*A3I*J^4*A2*A1)*PibarGammabarGammaPdoublecosetreplist[3]*Pfixerlist[22];
replist[39] - z^4*(J^8*A1*A3*A3*A3*A1*A3I*J^4*A2*A1)*PibarGammabarGammaPdoublecosetreplist[21]*Pfixerlist[11];
replist[40] - z^5*(J^8*A1*A3*A3*A3*A1*A2I*J^8*A3*A3*J^8)*PibarGammabarGammaPdoublecosetreplist[6]*Pfixerlist[11];
replist[41] - z*(J^8*A1*A3*A3*A3*A1*A2I*J^4)*PibarGammabarGammaPdoublecosetreplist[32]*Pfixerlist[19];
replist[42] - (1/z^2)*(A1I*A2)*PibarGammabarGammaPdoublecosetreplist[10]*Pfixerlist[18];
replist[43] - z^2*(A1I*A2I*A1*A3*A1I)*PibarGammabarGammaPdoublecosetreplist[20]*Pfixerlist[4];
replist[44] - (ID3)*PibarGammabarGammaPdoublecosetreplist[27]*Pfixerlist[24];
replist[45] - z^3*(J^8*A1*A3I*J^8*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[19]*Pfixerlist[19];
replist[46] - (J^8*A1*J^4*A2*A3I*A1I*A2)*PibarGammabarGammaPdoublecosetreplist[23]*Pfixerlist[9];
replist[47] - z^3*(J^8*A1*J^4*A2*A2*A1)*PibarGammabarGammaPdoublecosetreplist[33]*Pfixerlist[5];
replist[48] - (1/z^5)*(J^8*A3I*A3I*A3I*A2*A2*J^4*A3*A1*A3I*A1I*A2)*PibarGammabarGammaPdoublecosetreplist[23]*Pfixerlist[24];
replist[49] - (J^8*A3I*A3I*A3I*A2*A2*A1I*J^4*A2)*PibarGammabarGammaPdoublecosetreplist[8]*Pfixerlist[7];
replist[50] - (1/z^4)*(J^8*A1*A3*A3*A3*A1*A3I*J^4)*PibarGammabarGammaPdoublecosetreplist[9]*Pfixerlist[1];
replist[51] - (1/z^3)*(J^8*A1*A3*A3*A3*A1*A3I*J^4*A3*J^8*A1I*A2I*J^4)*PibarGammabarGammaPdoublecosetreplist[34]*Pfixerlist[1];
replist[52] - z^3*(J^8*A2*A1*J^8*A3*A3*A3*A1*J^4*A2*A2*A2*A1*J^4)*PibarGammabarGammaPdoublecosetreplist[20]*Pfixerlist[24];
replist[53] - (-1)*(A3*A3*A3*A2*A1*A2I*A1*J^8*A2*A1*J^4)*PibarGammabarGammaPdoublecosetreplist[29]*Pfixerlist[1];
replist[54] - (1/z^5)*(A1I*A2*A1I*A2I*J^4*A3*J^8)*PibarGammabarGammaPdoublecosetreplist[12]*Pfixerlist[9];
// From this we read off the first row of the table in Proposition 7. For example, PibarGammabarGammaPdoublecosetreplist[19]
// occurs 3 times on the right in the above 54 expressions, and so $m_{19}(M_c)=3$.


// From the set of 54 representatives for the double cosets 
// $\Pi_c\gamma(\bar\Gamma_c\cap\bar\Gamma_P)$ in
// $\Pi_c\backslash\bar\Gamma_c/(\bar\Gamma_c\cap\bar\Gamma_P)$,
// we get a set of 54 representatives $\gamma'$ for the double cosets 
// $\Pi_{-c}\gamma(\bar\Gamma_{-c}\cap k_{-c}\bar\Gamma_Pk_{-c}^{-1})$ in
// $\Pi_{-c}\backslash\bar\Gamma_{-c}/(\bar\Gamma_{-c}\capk_{-c}\bar\Gamma_Pk_{-c}^{-1})$,
// simply be conjugating by $k_{-c}$. According to Proposition 7, we need to find the
// images in $\Pi\backslash\bar\Gamma/\bar\Gamma_P$ of $\gamma'k_{-c}$ for each
// of these representatives $\gamma'$. That is, we need to find the image in $\Pi\backslash\bar\Gamma/\bar\Gamma_P$
// of $k_{-c}\gamma$ for each of the original 54 representatives $\gamma$. This we now do:

// The following are zero:
kmmm*replist[1] - (ID3)*PibarGammabarGammaPdoublecosetreplist[4]*Pfixerlist[1];
kmmm*replist[2] - z^2*(A2)*PibarGammabarGammaPdoublecosetreplist[18]*Pfixerlist[17];
kmmm*replist[3] - (J^4*A1I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[29]*Pfixerlist[10];
kmmm*replist[4] - (A2*A2)*PibarGammabarGammaPdoublecosetreplist[18]*Pfixerlist[1];
kmmm*replist[5] - (1/z^3)*(J^8*A3I*A3I*A1I*J^4)*PibarGammabarGammaPdoublecosetreplist[25]*Pfixerlist[9];
kmmm*replist[6] - z^2*(A2*A1I)*PibarGammabarGammaPdoublecosetreplist[8]*Pfixerlist[12];
kmmm*replist[7] - (1/z^3)*(A1I*A1I*A3I*A3I*A3I*J^4*A1I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[36]*Pfixerlist[10];
kmmm*replist[8] - (1/z^4)*(A1I*A3I*A3I*A3I*A2*A2)*PibarGammabarGammaPdoublecosetreplist[10]*Pfixerlist[22];
kmmm*replist[9] - z^3*(A3*A3*A3*A1*A1*A2)*PibarGammabarGammaPdoublecosetreplist[18]*Pfixerlist[5];
kmmm*replist[10] - (A2*A2*A1I)*PibarGammabarGammaPdoublecosetreplist[2]*Pfixerlist[7];
kmmm*replist[11] - (1/z^4)*(J^8*A1I*A3I*A3I*A3I*A2*A2*J^4)*PibarGammabarGammaPdoublecosetreplist[12]*Pfixerlist[2];
kmmm*replist[12] - (1/z^3)*(J^8*A3I*A3I*A1I*J^4)*PibarGammabarGammaPdoublecosetreplist[13]*Pfixerlist[15];
kmmm*replist[13] - (1/z)*(J^8*A3I*A3I*A1I*J^4)*PibarGammabarGammaPdoublecosetreplist[1]*Pfixerlist[14];
kmmm*replist[14] - z*(A2*A1I*A2I)*PibarGammabarGammaPdoublecosetreplist[1]*Pfixerlist[21];
kmmm*replist[15] - z*(J^8*A1*A2*A2*J^4*A1)*PibarGammabarGammaPdoublecosetreplist[11]*Pfixerlist[6];
kmmm*replist[16] - (A2*J^4*A2*A1*A2I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[14]*Pfixerlist[1];
kmmm*replist[17] - z^3*(A2*A1I*A2I)*PibarGammabarGammaPdoublecosetreplist[10]*Pfixerlist[20];
kmmm*replist[18] - z^2*(J^4*A1I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[15]*Pfixerlist[19];
kmmm*replist[19] - (A3I)*PibarGammabarGammaPdoublecosetreplist[16]*Pfixerlist[12];
kmmm*replist[20] - (1/z^4)*(A1I*A3I*A3I*A3I*A2)*PibarGammabarGammaPdoublecosetreplist[4]*Pfixerlist[8];
kmmm*replist[21] - (J^8*A3I*A1*A2*A2*J^4)*PibarGammabarGammaPdoublecosetreplist[32]*Pfixerlist[1];
kmmm*replist[22] - (A2*A2*A1I)*PibarGammabarGammaPdoublecosetreplist[26]*Pfixerlist[13];
kmmm*replist[23] - (1/z^4)*(J^8*A1I*A3I*A3I*A3I*A2*J^4)*PibarGammabarGammaPdoublecosetreplist[6]*Pfixerlist[8];
kmmm*replist[24] - z*(A1I*A2I)*PibarGammabarGammaPdoublecosetreplist[28]*Pfixerlist[18];
kmmm*replist[25] - z^2*(A2*A1*J^4*A2I*A1*A3*A2I*A2I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[2]*Pfixerlist[6];
kmmm*replist[26] - (1/z^5)*(J^8*A3I*A3I*A1I*A3I*J^4*A1I*A3I*A3I*A3I)*PibarGammabarGammaPdoublecosetreplist[13]*Pfixerlist[22];
kmmm*replist[27] - (A2*A2)*PibarGammabarGammaPdoublecosetreplist[17]*Pfixerlist[24];
kmmm*replist[28] - (1/z^2)*(J^8*A3I*A2*A1*J^4*A3I*A1I*A2*A1)*PibarGammabarGammaPdoublecosetreplist[33]*Pfixerlist[22];
kmmm*replist[29] - (1/z)*(A1I*A1I*A3I*A3I*A3I*J^4*A1I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[9]*Pfixerlist[19];
kmmm*replist[30] - (1/z^3)*(A1I*A1I*A3I*A3I*A3I*A2*A1)*PibarGammabarGammaPdoublecosetreplist[9]*Pfixerlist[17];
kmmm*replist[31] - (1/z^4)*(J^8*A1*A2*A2*A3I*A1I*A2*J^8*A1I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[14]*Pfixerlist[5];
kmmm*replist[32] - z^2*(A2*A1I*A2I*A2I*A2I)*PibarGammabarGammaPdoublecosetreplist[13]*Pfixerlist[5];
kmmm*replist[33] - (-1)*(A2*A1I*J^4*A2*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[35]*Pfixerlist[18];
kmmm*replist[34] - (1/z)*(A1I*A3I*A3I*A3I*J^8*A1*J^4)*PibarGammabarGammaPdoublecosetreplist[7]*Pfixerlist[21];
kmmm*replist[35] - (1/z)*(J^4*A1I*A2I*A3I*A3I*J^8)*PibarGammabarGammaPdoublecosetreplist[12]*Pfixerlist[6];
kmmm*replist[36] - z^3*(A3I*J^8*A2*A1*A2I*A2I*J^4)*PibarGammabarGammaPdoublecosetreplist[15]*Pfixerlist[2];
kmmm*replist[37] - (ID3)*PibarGammabarGammaPdoublecosetreplist[5]*Pfixerlist[1];
kmmm*replist[38] - (1/z^3)*(J^4*A1I*A1I*A3I*A3I*A3I*J^4*A1I*A2I*J^4)*PibarGammabarGammaPdoublecosetreplist[34]*Pfixerlist[10];
kmmm*replist[39] - z*(J^8*A3I*A1*A2*A2*A1I*A2I*J^8*A1I*J^8)*PibarGammabarGammaPdoublecosetreplist[3]*Pfixerlist[6];
kmmm*replist[40] - z^2*(A2*A2*A1I)*PibarGammabarGammaPdoublecosetreplist[5]*Pfixerlist[15];
kmmm*replist[41] - (1/z)*(A2*A2*A1I)*PibarGammabarGammaPdoublecosetreplist[14]*Pfixerlist[15];
kmmm*replist[42] - (1/z^4)*(J^8*A1I*A3I*A3I*A3I*A2*J^4)*PibarGammabarGammaPdoublecosetreplist[3]*Pfixerlist[14];
kmmm*replist[43] - z^3*(J^8*A3*A3*A3*A1*A1*A2*J^4)*PibarGammabarGammaPdoublecosetreplist[17]*Pfixerlist[5];
kmmm*replist[44] - (ID3)*PibarGammabarGammaPdoublecosetreplist[6]*Pfixerlist[24];
kmmm*replist[45] - z^3*(A2*A1*J^4*A2I*A1*A3*A2I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[16]*Pfixerlist[20];
kmmm*replist[46] - (1/z^3)*(J^8*A3I*A3I*A1I*J^4*A3*A3*A3*A1*J^4*A2*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[31]*Pfixerlist[24];
kmmm*replist[47] - (1/z^4)*(A3*A3*A3*A1*A1*A2*A2*A2)*PibarGammabarGammaPdoublecosetreplist[15]*Pfixerlist[8];
kmmm*replist[48] - (1/z^2)*(J^8*A3I*A3I*A1I*A3I*J^4)*PibarGammabarGammaPdoublecosetreplist[11]*Pfixerlist[2];
kmmm*replist[49] - (1/z^4)*(J^8*A3I*A3I*A2I*A1I*J^4)*PibarGammabarGammaPdoublecosetreplist[7]*Pfixerlist[11];
kmmm*replist[50] - z^3*(J^8*A3I*A2*A1*J^4*A3*A3*A1*J^4*A2*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[16]*Pfixerlist[22];
kmmm*replist[51] - (1/z^2)*(A2*A1I*A2I*J^8*A1*A3I*J^4)*PibarGammabarGammaPdoublecosetreplist[8]*Pfixerlist[5];
kmmm*replist[52] - z^2*(A2*A1I*A2I*A2I*A2I)*PibarGammabarGammaPdoublecosetreplist[30]*Pfixerlist[18];
kmmm*replist[53] - (1/z^2)*(A2*A1I*A3I*J^4*A1I*A3I*A3I*A3I*A2*A2*J^8*A1I)*PibarGammabarGammaPdoublecosetreplist[17]*Pfixerlist[12];
kmmm*replist[54] - (1/z^5)*(J^8*A1I*A3I*A3I*A3I*A2*J^4)*PibarGammabarGammaPdoublecosetreplist[27]*Pfixerlist[13];
// From this we read off the second row of the table in Proposition 7. For example, PibarGammabarGammaPdoublecosetreplist[13]
// occurs 3 times on the right in the above 54 expressions, and so $m_{13}(M_{-c})=3$.

t2:=(B*U^-1)^2;
t4:=J^-1*t2;
t12:=J;
zc:=U;
// The following is a list of 18 representatives of the double cosets $b^{-1}\Pi_{b(M_c)}b\gamma(\bar\Gamma_c\cap\bar\Gamma_P)$
// in $b^{-1}\Pi_{b(M_c)}b\backslash\bar\Gamma_c/(\bar\Gamma_c\cap\bar\Gamma_P)$, thought of as matrices.

replist:=[ MatsL |
ID3,t4,t12,t4^-1,t2*t4,t4^2,t4*t12,t12^2,t12*t4^-1,t12^-1*t4,
t4^2*t12,t4*t12^2,t12*t2*t4,t12^3,t4^-1*t2*t4,t12^-2*t4,t12*t2*t12^-1*t4,t12^3*t4^-1];

// So the elements $B*\gamma*BI$ for the $\gamma$'s in this list form a set of
// representatives $\gamma'$ for the double cosets $\Pi_{b(M_c)}b\gamma'b(\bar\Gamma_c\cap\bar\Gamma_P)b^{-1}$ in 
// $\Pi_{b(M_c)}\backslash\bar\Gamma_{b(M_c)}/b(\bar\Gamma_c\cap\bar\Gamma_P)b^{-1}$. 
// To get the third row of the tabel in Proposition 7, for each of these $\gamma'$'s, we need to 
// project $\gamma'*B$ onto $\Pi\backslash\bar\Gamma/\bar\Gamma_P$.
// So we need to project $B*\gamma$, for each of the $\gamma$'s in replist, 
// onto $\Pi\backslash\bar\Gamma/\bar\Gamma_P$.


B*replist[1] - (ID3)*PibarGammabarGammaPdoublecosetreplist[25]*Pfixerlist[4];
B*replist[2] - z*(A2*A2*A2)*PibarGammabarGammaPdoublecosetreplist[29]*Pfixerlist[6];
B*replist[3] - (ID3)*PibarGammabarGammaPdoublecosetreplist[34]*Pfixerlist[13];
B*replist[4] - z^2*(J^8*A1*A3I*J^8*A2*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[35]*Pfixerlist[7];
B*replist[5] - z*(J^4*A1I*J^8)*PibarGammabarGammaPdoublecosetreplist[21]*Pfixerlist[7];
B*replist[6] - z*(J^8*A2*A1*A2I*A2I*A1I*A3I*J^4*A2)*PibarGammabarGammaPdoublecosetreplist[33]*Pfixerlist[11];
B*replist[7] - z^3*(A3*A3*A3*A1*A2*A1)*PibarGammabarGammaPdoublecosetreplist[24]*Pfixerlist[12];
B*replist[8] - (J^8*A1*A2*A2*J^4)*PibarGammabarGammaPdoublecosetreplist[32]*Pfixerlist[12];
B*replist[9] - (J^8*A1*J^4)*PibarGammabarGammaPdoublecosetreplist[23]*Pfixerlist[17];
B*replist[10] - z^5*(A3*A3*A3*A1)*PibarGammabarGammaPdoublecosetreplist[31]*Pfixerlist[17];
B*replist[11] - z^2*(J^4*A2I*A2I*A1I*A3I*J^8)*PibarGammabarGammaPdoublecosetreplist[28]*Pfixerlist[19];
B*replist[12] - z*(A3*A3*A3*A1*A2*J^8*A1I*J^4)*PibarGammabarGammaPdoublecosetreplist[19]*Pfixerlist[9];
B*replist[13] - (-1)*(J^8*A1I*A3I*A3I*A3I*A2*A2*J^8*A1I*J^8)*PibarGammabarGammaPdoublecosetreplist[27]*Pfixerlist[6];
B*replist[14] - (J^8*A1*A2*A2*A1I*A2I*J^4)*PibarGammabarGammaPdoublecosetreplist[30]*Pfixerlist[20];
B*replist[15] - z^4*(J^8*A1*A3I*J^8*A2*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[26]*Pfixerlist[19];
B*replist[16] - z^5*(A3*A3*A3*A1*A3*A3*J^4*A1I*J^8)*PibarGammabarGammaPdoublecosetreplist[36]*Pfixerlist[9];
B*replist[17] - (1/z^4)*(J^8*A1I*A3I*A3I*A3I*A2*A2*A1*J^4)*PibarGammabarGammaPdoublecosetreplist[20]*Pfixerlist[9];
B*replist[18] - (1/z)*(J^8*A1*A2*A2*J^8*A2*A1*J^8)*PibarGammabarGammaPdoublecosetreplist[22]*Pfixerlist[12];
// We see that $PibarGammabarGammaPdoublecosetreplist[i]$, for $i=19,\ldots,36$ appears just once on the right 
// in the 18 expressions above, and so $m_i(M)=1$ for $M=b(M_c)$ for these $i$'s, and $m_i(M)=0$ for this~$M$
// and $i=1,\ldots,18$.

// The following is a list of 18 representatives of the double cosets 
// $b\Pi_{b^{-1}(M_c)}b^{-1}\gamma(\bar\Gamma_c\cap\bar\Gamma_P)$
// in $b\Pi_{b^{-1}(M_c)}b^{-1}\backslash\bar\Gamma_c/(\bar\Gamma_c\cap\bar\Gamma_P)$, thought of as matrices.

replist:=[ MatsL  | ID3,t4,t12,t4^-1,t4^2,t4*t12,t12*t4^-1,
t4^-1*t12,t12^-1*t4,t4^2*t12,t4*t12^2,t4*t12*t4^-1,t4*t12^-1*t4,t4^-1*t2*t4,
t4^-1*t12^2,t12^-1*t4*t12,(t4*t12*t4^-1)^2,t4*t12^-1*t4*t12];

BI*replist[1] - z*(ID3)*PibarGammabarGammaPdoublecosetreplist[25]*Pfixerlist[17];
BI*replist[2] - z^2*(ID3)*PibarGammabarGammaPdoublecosetreplist[31]*Pfixerlist[14];
BI*replist[3] - (A2*A2*A2)*PibarGammabarGammaPdoublecosetreplist[15]*Pfixerlist[10];
BI*replist[4] - (1/z)*(J^8*A2*A1*J^4)*PibarGammabarGammaPdoublecosetreplist[32]*Pfixerlist[9];
BI*replist[5] - z^3*(ID3)*PibarGammabarGammaPdoublecosetreplist[16]*Pfixerlist[15];
BI*replist[6] - z*(J^4*A1I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[18]*Pfixerlist[15];
BI*replist[7] - (1/z^3)*(J^8*A3I*A3I*A3I*A2*A2*A2*J^4)*PibarGammabarGammaPdoublecosetreplist[28]*Pfixerlist[8];
BI*replist[8] - (1/z)*(J^8*A2*A1*J^4)*PibarGammabarGammaPdoublecosetreplist[26]*Pfixerlist[16];
BI*replist[9] - z^3*(J^4*A1I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[27]*Pfixerlist[18];
BI*replist[10] - z^2*(A3)*PibarGammabarGammaPdoublecosetreplist[35]*Pfixerlist[19];
BI*replist[11] - (1/z^3)*(A1I*A3I*A3I*A3I*A2*A2)*PibarGammabarGammaPdoublecosetreplist[34]*Pfixerlist[19];
BI*replist[12] - (J^4*A1I*A2I*J^8)*PibarGammabarGammaPdoublecosetreplist[33]*Pfixerlist[14];
BI*replist[13] - z*(J^4*A2I*A1I*J^8)*PibarGammabarGammaPdoublecosetreplist[36]*Pfixerlist[5];
BI*replist[14] - z*(J^8*A2*A1*J^4)*PibarGammabarGammaPdoublecosetreplist[17]*Pfixerlist[14];
BI*replist[15] - (1/z^5)*(J^8*A2*A1*J^4*A3I*A3I*A3I)*PibarGammabarGammaPdoublecosetreplist[13]*Pfixerlist[12];
BI*replist[16] - (1/z)*(J^4*A1I*A2I*A3I*A3I*A3I*J^8)*PibarGammabarGammaPdoublecosetreplist[14]*Pfixerlist[11];
BI*replist[17] - (1/z^4)*(A1I*A3I*A3I*A3I*A2*A2)*PibarGammabarGammaPdoublecosetreplist[30]*Pfixerlist[14];
BI*replist[18] - z*(J^4*A2I*A1I*J^8)*PibarGammabarGammaPdoublecosetreplist[29]*Pfixerlist[14];

// We see that $PibarGammabarGammaPdoublecosetreplist[i]$, for $i=13,\ldots,18$ and for $i=25,\ldots,36$
// appears just once on the right in the 18 expressions above, and so $m_i(M)=1$ for $M=b^{-1}(M_c)$ for 
// these $i$'s, and $m_i(M)=0$ for this~$M$ and for the other $i$'s between~1 and~36.

// End of calculations for Proposition 7.
// **************************************

// Calculations for Proposition 8.
// *******************************

gamma12:=B*V;

// The fixed point $\xi_{12}$ in $B(\C^2)$ of $\gamma_{12}$ is $(0,\zeta-1)$.
// We can check this by calculating the following two expressions:
gamma12[1][2]*(z-1)+gamma12[1][3];                                       // Output: 0
(gamma12[2][2]*(z-1)+gamma12[2][3])/(gamma12[3][2]*(z-1)+gamma12[3][3]); // Output: z-1

// Here is a list of 72 elements of $\bar\Gamma$. In the file cs-surface-presentations_magma.txt,
// we check that they form a complete set of representatives for the double cosets
// $\Pi g\langle\gamma_{12}\rangle$, $g\in\bar\Gamma$.
// This list is ordered so that for $i=1,\ldots,24$, $\Pi(t_i\xi_{12})$ is in the image
// of $M_0$, under the natural map $B(\C^2)\to\Pi\backslash B(\C^2)$, and so that, similarly,
// for $i=25,\ldots,48$, $\Pi(t_i\xi_{12})$ is in the image of $M_1$, and 
// for $i=49,\ldots,72$, $\Pi(t_i\xi_{12})$ is in the image of $M_\infty$, as we check below.

gamma12doublecosetlist:=[ MatsL  |
J^2,J^6,J^10,
U*J,U*J^5,U*J^9,
ID3,J^4,J^8,
U*J^3,U*J^7,U*J^11,
J,J^5,J^9,
U*J^2,U*J^6,U*J^10,
J^3,J^7,J^11,
U,U*J^4,U*J^8,
//
VI*U*J,VI*U*J^5,VI*U*J^9,
VI*U*J^3,VI*U*J^7,VI*U*J^11,
B*U,B*U*J^4,B*U*J^8,
VI*U*J^2,VI*U*J^6,VI*U*J^10,
B*U*J,B*U*J^5,B*U*J^9,
B*V*U*J,B*V*U*J^5,B*V*U*J^9,
B*VI*U*J,B*VI*U*J^5,B*VI*U*J^9,
VI*U,VI*U*J^4,VI*U*J^8,
//
V*UI*J,V*UI*J^5,V*UI*J^9,
V*UI*J^2,V*UI*J^6,V*UI*J^10,
U*VI*U*J,U*VI*U*J^5,U*VI*U*J^9,
B^-1*V*U*J^2,B^-1*V*U*J^6,B^-1*V*U*J^10,
V*UI,V*UI*J^4,V*UI*J^8,
V*UI*J^3,V*UI*J^7,V*UI*J^11,
U*VI*U,U*VI*U*J^4,U*VI*U*J^8,
B*V*UI*V*V*U*J,B*V*UI*V*V*U*J^5,B*V*UI*V*V*U*J^9];

gamma12doublecosetwordlist:=[
["J^2"],["J^6"],["J^10"],
["U*J"],["U*J^5"],["U*J^9"],
["ID3"],["J^4"],["J^8"],
["U*J^3"],["U*J^7"],["U*J^11"],
["J"],["J^5"],["J^9"],
["U*J^2"],["U*J^6"],["U*J^10"],
["J^3"],["J^7"],["J^11"],
["U"],["U*J^4"],["U*J^8"],
//
["VI*U*J"],["VI*U*J^5"],["VI*U*J^9"],
["VI*U*J^3"],["VI*U*J^7"],["VI*U*J^11"],
["B*U"],["B*U*J^4"],["B*U*J^8"],
["VI*U*J^2"],["VI*U*J^6"],["VI*U*J^10"],
["B*U*J"],["B*U*J^5"],["B*U*J^9"],
["B*V*U*J"],["B*V*U*J^5"],["B*V*U*J^9"],
["B*VI*U*J"],["B*VI*U*J^5"],["B*VI*U*J^9"],
["VI*U"],["VI*U*J^4"],["VI*U*J^8"],
//
["V*UI*J"],["V*UI*J^5"],["V*UI*J^9"],
["V*UI*J^2"],["V*UI*J^6"],["V*UI*J^10"],
["U*VI*U*J"],["U*VI*U*J^5"],["U*VI*U*J^9"],
["B^-1*V*U*J^2"],["B^-1*V*U*J^6"],["B^-1*V*U*J^10"],
["V*UI"],["V*UI*J^4"],["V*UI*J^8"],
["V*UI*J^3"],["V*UI*J^7"],["V*UI*J^11"],
["U*VI*U"],["U*VI*U*J^4"],["U*VI*U*J^8"],
["B*V*UI*V*V*U*J"],["B*V*UI*V*V*U*J^5"],["B*V*UI*V*V*U*J^9"]];

// Here is a list of 72 elements $\pi_i$ of~$\Pi$. They were
// chosen so that: 
// For $i=1,\ldots,24$, $\pi_it_i\xi_{12}\in M_0$.
// For $i=25,\ldots,48$, $\pi_it_i\xi_{12}\in M_1$.
// For $i=49,\ldots,72$, $\pi_it_i\xi_{12}\in M_\infty$.
pilist1:=[MatsL |
ID3,
ID3,
ID3,
A1I*A2I*A1,
A1I*A2I*A1,
A1I*A2I*A1,
ID3,
ID3,
ID3,
A1I*A2I*A1,
A1I*A2I*A1,
A1I*A2I*A1,
ID3,
ID3,
ID3,
A1I*A2I*A1,
A1I*A2I*A1,
A1I*A2I*A1,
ID3,
ID3,
ID3,
A1I*A2I*A1,
A1I*A2I*A1,
A1I*A2I*A1,
ID3,
ID3,
ID3,
ID3,
ID3,
ID3,
A1I*A3I*A3I*A3I,
A1I*A3I*A3I*A3I,
A1I*A3I*A3I*A3I,
ID3,
ID3,
ID3,
A1I*A3I*A3I*A3I,
A1I*A3I*A3I*A3I,
A1I*A3I*A3I*A3I,
A3*A3*A3*A1*A1*A2I,
A3*A3*A3*A1*A1*A2I,
A3*A3*A3*A1*A1*A2I,
A2I*A2I,
A2I*A2I,
A2I*A2I,
ID3,
ID3,
ID3,
A2I,
A2I,
A2I,
A2I,
A2I,
A2I,
ID3,
ID3,
ID3,
A3I*A1I*A2I*A2I,
A3I*A1I*A2I*A2I,
A3I*A1I*A2I*A2I,
A2I,
A2I,
A2I,
A2I,
A2I,
A2I,
ID3,
ID3,
ID3,
A2I*A1*A2I,
A2I*A1*A2I,
A2I*A1*A2I];

print "The length of pilist1 is",#pilist1; // Output: 72

print "Checking that for i=1,...,24, pi_i*t_i, applied";
print "to the fixed point of gamma_{12}, is in M_0.";

for ii in [1 .. 24] do
 tmpmat:=pilist1[ii]*gamma12doublecosetlist[ii];
 print tmpmat[1][2]*(z-1)+tmpmat[1][3];
end for;

print "Checking that for i=25,...,48, pi_i*t_i, applied";
print "to the fixed point of gamma_{12}, is in M_1.";
for ii in [25 .. 48] do
 tmpmat:=pilist1[ii]*gamma12doublecosetlist[ii];
 print (tmpmat[1][2]*(z-1)+tmpmat[1][3]) - (tmpmat[2][2]*(z-1)+tmpmat[2][3]);
end for;

print "Checking that for i=49,...,72, pi_i*t_i, applied";
print "to the fixed point of gamma_{12}, is in M_infty.";
for ii in [49 .. 72] do
 tmpmat:=pilist1[ii]*gamma12doublecosetlist[ii];
 print tmpmat[2][2]*(z-1)+tmpmat[2][3];
end for;

// The following is a list of 72 elements $\pi_i'$ of $\Pi$ such that
// if $t_i$ is the $i$-th element of gamma12doublecosetlist, then
// $\pi_i't_i\xi_{12}$ is in one of $M_c$, $M_{-c}$, $b(M_c)$ and $b^{-1}(M_c)$:

pilist2:=[MatsL |
A1I*A2I*A2I*A1I*A1I*A2I,
A1*A3I*A3I*A1I*A2,
A1*A3I*A1*A3*A1,
A2I*A2I*A2I*A3*A3,
A1*A3I*A1I,
A1I*A2I*A1*A3,
A1I*A3I*A3I*A3I*A2,
A2*A2*A1I,
A2*A1I*A2*A2,
A1I*A2I,
A2*A2*A1I*A3I*A2,
A2*A1I*A2I*A2I*A2I,
A3*A3*A3*A1,
A2*A2*A2*A1I*A3I,
A2*A1I*A1I*A3I*A1*A3I,
A3*A3*A3*A1*A3*A3,
A2*A1I*A1I*A3I*A1*A3I*A1I,
A2*A2*A2*A3I,
A3,
A2*A2*A1*A3*A2*A2*A2,
A3*A1,
ID3,
A2*A2*A1*A3*A1I,
A3*A1*A2,
A1*A3I*A2I*A1*A3I,
A1*A3I*A1*A3*A1*A3*A1I,
A1I*A2,
A1*A3I*A1*A3*A2*A3I,
A1*A3I*A3I*A1I*A2*A3I*A1I,
A1I*A2I*A2I,
ID3,
A1I*A2*A2*A1*A2I,
A1I*A2I*A1*A3*A2I*A2I,
A3I,
A1*A2*A2*A1I*A2I*A1*A3*A1I,
A2,
A1I*A2I*A1*A2*A2,
A2*A2*A1I*A3I*A1*A2I,
A2I*A2I,
A2,
A2*A2*A1I*A3I*A2*A1*A2I*A2I,
A1I*A2I*A3I*A3I*A1*A2I*A2I*A2I,
A2*A2*A2*A3I*A1I*A3I*A3I*A3I,
A2*A1I*A3I*A3I*A3I*A1I*A3I*A3I*A3I,
A2*A2*A2*A3I*A2I,
A2*A2*A1*A3,
A3*A1*A3*A3*A3,
A2*A2*A2,
A1I*A2I,
A1*A3I*A2I*A1*A3,
A1I*A2I*A2I*A1I*A2*A2*A1,
A1I*A2I*A2I*A1I*A2,
A1I*A2I*A2I*A1I*A3I,
A2*A1,
A3*A2I*A2I*A2I*A3*A3*A3*A1,
A1*A3I*A2I,
A1,
A1*A3I*A1*A3*A2I*A2I,
A1I*A2*A2*A3*A1*A3I*A1I*A2I*A2I,
A1*A3I*A2I*A1*A3I,
A2*A1I*A2I,
A1I*A2I*A3I*A3I,
A2*A1I*A1I*A3I*A3I*A3I*A2*A2*A1,
A2*A2*A1I*A3*A1,
A1I*A2I*A3I*A3I*A1*A3I,
A1I*A2I*A1*A2*A2,
A1*A2*A1I*A1I*A3I*A3I*A3I*A2*A1,
A2*A2*A1I*A3*A1*A2I,
A1I*A2I*A3I*A3I*A1,
A2*A1I*A2I*A2I,
A2*A1I*A2I*A2I*A2I*A3I,
A1*A3*A2I*A2I*A2I*A3*A3*A3*A1];

piwordlist2:=[
["A1I*A2I*A2I*A1I*A1I*A2I"],
["A1*A3I*A3I*A1I*A2"],
["A1*A3I*A1*A3*A1"],
["A2I*A2I*A2I*A3*A3"],
["A1*A3I*A1I"],
["A1I*A2I*A1*A3"],
["A1I*A3I*A3I*A3I*A2"],
["A2*A2*A1I"],
["A2*A1I*A2*A2"],
["A1I*A2I"],
["A2*A2*A1I*A3I*A2"],
["A2*A1I*A2I*A2I*A2I"],
["A3*A3*A3*A1"],
["A2*A2*A2*A1I*A3I"],
["A2*A1I*A1I*A3I*A1*A3I"],
["A3*A3*A3*A1*A3*A3"],
["A2*A1I*A1I*A3I*A1*A3I*A1I"],
["A2*A2*A2*A3I"],
["A3"],
["A2*A2*A1*A3*A2*A2*A2"],
["A3*A1"],
["ID3"],
["A2*A2*A1*A3*A1I"],
["A3*A1*A2"],
["A1*A3I*A2I*A1*A3I"],
["A1*A3I*A1*A3*A1*A3*A1I"],
["A1I*A2"],
["A1*A3I*A1*A3*A2*A3I"],
["A1*A3I*A3I*A1I*A2*A3I*A1I"],
["A1I*A2I*A2I"],
["ID3"],
["A1I*A2*A2*A1*A2I"],
["A1I*A2I*A1*A3*A2I*A2I"],
["A3I"],
["A1*A2*A2*A1I*A2I*A1*A3*A1I"],
["A2"],
["A1I*A2I*A1*A2*A2"],
["A2*A2*A1I*A3I*A1*A2I"],
["A2I*A2I"],
["A2"],
["A2*A2*A1I*A3I*A2*A1*A2I*A2I"],
["A1I*A2I*A3I*A3I*A1*A2I*A2I*A2I"],
["A2*A2*A2*A3I*A1I*A3I*A3I*A3I"],
["A2*A1I*A3I*A3I*A3I*A1I*A3I*A3I*A3I"],
["A2*A2*A2*A3I*A2I"],
["A2*A2*A1*A3"],
["A3*A1*A3*A3*A3"],
["A2*A2*A2"],
["A1I*A2I"],
["A1*A3I*A2I*A1*A3"],
["A1I*A2I*A2I*A1I*A2*A2*A1"],
["A1I*A2I*A2I*A1I*A2"],
["A1I*A2I*A2I*A1I*A3I"],
["A2*A1"],
["A3*A2I*A2I*A2I*A3*A3*A3*A1"],
["A1*A3I*A2I"],
["A1"],
["A1*A3I*A1*A3*A2I*A2I"],
["A1I*A2*A2*A3*A1*A3I*A1I*A2I*A2I"],
["A1*A3I*A2I*A1*A3I"],
["A2*A1I*A2I"],
["A1I*A2I*A3I*A3I"],
["A2*A1I*A1I*A3I*A3I*A3I*A2*A2*A1"],
["A2*A2*A1I*A3*A1"],
["A1I*A2I*A3I*A3I*A1*A3I"],
["A1I*A2I*A1*A2*A2"],
["A1*A2*A1I*A1I*A3I*A3I*A3I*A2*A1"],
["A2*A2*A1I*A3*A1*A2I"],
["A1I*A2I*A3I*A3I*A1"],
["A2*A1I*A2I*A2I"],
["A2*A1I*A2I*A2I*A2I*A3I"],
["A1*A3*A2I*A2I*A2I*A3*A3*A3*A1"]];

// Checking that pilist2 has the stated property:
// for ii in [1 .. 72] do
//   testmat:=pilist2[ii]*gamma12doublecosetlist[ii];
//   testquant:=testmat[1][2]*(z-1)+testmat[1][3] - c*(testmat[2][2]*(z-1)+testmat[2][3]);
//   if (testquant eq 0) then print "For ii = ",ii,piwordlist2[ii],"gives a pi such that pi*t_i*xi_12 isnin M_c"; end if;
//   testquant:=testmat[1][2]*(z-1)+testmat[1][3] - (-c)*(testmat[2][2]*(z-1)+testmat[2][3]);
//   if (testquant eq 0) then print "For ii = ",ii,piwordlist2[ii],"gives a pi such that pi*t_i*xi_12 is in M_(-c)"; end if;
//   testmat:=BI*pilist2[ii]*gamma12doublecosetlist[ii];
//   testquant:=testmat[1][2]*(z-1)+testmat[1][3] - c*(testmat[2][2]*(z-1)+testmat[2][3]);
//   if (testquant eq 0) then print "For ii = ",ii,piwordlist2[ii],"gives a pi such that pi*t_i*xi_12 is in b(M_c)"; end if;
//   testmat:=B*pilist2[ii]*gamma12doublecosetlist[ii];
//   testquant:=testmat[1][2]*(z-1)+testmat[1][3] - c*(testmat[2][2]*(z-1)+testmat[2][3]);
//   if (testquant eq 0) then print "For ii = ",ii,piwordlist2[ii],"gives a pi such that pi*t_i*xi_12 is in b^-1(M_c)"; end if;
// end for;

// For ii =  1, A1I*A2I*A2I*A1I*A1I*A2I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  2, A1*A3I*A3I*A1I*A2 gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  3, A1*A3I*A1*A3*A1 gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  4, A2I*A2I*A2I*A3*A3 gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  5, A1*A3I*A1I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  6, A1I*A2I*A1*A3 gives a pi such that pi*t_i*xi_12 is in M_c
// 
// For ii =  7, A1I*A3I*A3I*A3I*A2 gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  8, A2*A2*A1I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  9, A2*A1I*A2*A2 gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  10, A1I*A2I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  11, A2*A2*A1I*A3I*A2 gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  12, A2*A1I*A2I*A2I*A2I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// 
// For ii =  13, A3*A3*A3*A1 gives a pi such that pi*t_i*xi_12 is in b(M_c)
// For ii =  14, A2*A2*A2*A1I*A3I gives a pi such that pi*t_i*xi_12 is in b(M_c)
// For ii =  15, A2*A1I*A1I*A3I*A1*A3I gives a pi such that pi*t_i*xi_12 is in b(M_c)
// For ii =  16, A3*A3*A3*A1*A3*A3 gives a pi such that pi*t_i*xi_12 is in b(M_c)
// For ii =  17, A2*A1I*A1I*A3I*A1*A3I*A1I gives a pi such that pi*t_i*xi_12 is in b(M_c)
// For ii =  18, A2*A2*A2*A3I gives a pi such that pi*t_i*xi_12 is in b(M_c)
// 
// For ii =  19, A3 gives a pi such that pi*t_i*xi_12 is in b^-1(M_c)
// For ii =  20, A2*A2*A1*A3*A2*A2*A2 gives a pi such that pi*t_i*xi_12 is in b^-1(M_c)
// For ii =  21, A3*A1 gives a pi such that pi*t_i*xi_12 is in b^-1(M_c)
// For ii =  22, ID3 gives a pi such that pi*t_i*xi_12 is in b^-1(M_c)
// For ii =  23, A2*A2*A1*A3*A1I gives a pi such that pi*t_i*xi_12 is in b^-1(M_c)
// For ii =  24, A3*A1*A2 gives a pi such that pi*t_i*xi_12 is in b^-1(M_c)
// 
// For ii =  25, A1*A3I*A2I*A1*A3I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  26, A1*A3I*A1*A3*A1*A3*A1I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  27, A1I*A2 gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  28, A1*A3I*A1*A3*A2*A3I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  29, A1*A3I*A3I*A1I*A2*A3I*A1I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  30, A1I*A2I*A2I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  31, ID3 gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  32, A1I*A2*A2*A1*A2I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  33, A1I*A2I*A1*A3*A2I*A2I gives a pi such that pi*t_i*xi_12 is in M_c
// 
// For ii =  34, A3I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  35, A1*A2*A2*A1I*A2I*A1*A3*A1I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  36, A2 gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  37, A1I*A2I*A1*A2*A2 gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  38, A2*A2*A1I*A3I*A1*A2I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  39, A2I*A2I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  40, A2 gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  41, A2*A2*A1I*A3I*A2*A1*A2I*A2I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  42, A1I*A2I*A3I*A3I*A1*A2I*A2I*A2I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// 
// For ii =  43, A2*A2*A2*A3I*A1I*A3I*A3I*A3I gives a pi such that pi*t_i*xi_12 is in b(M_c)
// For ii =  44, A2*A1I*A3I*A3I*A3I*A1I*A3I*A3I*A3I gives a pi such that pi*t_i*xi_12 is in b(M_c)
// For ii =  45, A2*A2*A2*A3I*A2I gives a pi such that pi*t_i*xi_12 is in b(M_c)
// 
// For ii =  46, A2*A2*A1*A3 gives a pi such that pi*t_i*xi_12 is in b^-1(M_c)
// For ii =  47, A3*A1*A3*A3*A3 gives a pi such that pi*t_i*xi_12 is in b^-1(M_c)
// For ii =  48, A2*A2*A2 gives a pi such that pi*t_i*xi_12 is in b^-1(M_c)
// 
// For ii =  49, A1I*A2I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  50, A1*A3I*A2I*A1*A3 gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  51, A1I*A2I*A2I*A1I*A2*A2*A1 gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  52, A1I*A2I*A2I*A1I*A2 gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  53, A1I*A2I*A2I*A1I*A3I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  54, A2*A1 gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  55, A3*A2I*A2I*A2I*A3*A3*A3*A1 gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  56, A1*A3I*A2I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  57, A1 gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  58, A1*A3I*A1*A3*A2I*A2I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  59, A1I*A2*A2*A3*A1*A3I*A1I*A2I*A2I gives a pi such that pi*t_i*xi_12 is in M_c
// For ii =  60, A1*A3I*A2I*A1*A3I gives a pi such that pi*t_i*xi_12 is in M_c
// 
// For ii =  61, A2*A1I*A2I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  62, A1I*A2I*A3I*A3I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  63, A2*A1I*A1I*A3I*A3I*A3I*A2*A2*A1 gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  64, A2*A2*A1I*A3*A1 gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  65, A1I*A2I*A3I*A3I*A1*A3I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  66, A1I*A2I*A1*A2*A2 gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  67, A1*A2*A1I*A1I*A3I*A3I*A3I*A2*A1 gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  68, A2*A2*A1I*A3*A1*A2I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  69, A1I*A2I*A3I*A3I*A1 gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  70, A2*A1I*A2I*A2I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  71, A2*A1I*A2I*A2I*A2I*A3I gives a pi such that pi*t_i*xi_12 is in M_(-c)
// For ii =  72, A1*A3*A2I*A2I*A2I*A3*A3*A3*A1 gives a pi such that pi*t_i*xi_12 is in M_(-c)

// Here is the routine used to find the list pilist2:

// Tlist:=gamma12doublecosetlist;
// Twordlist:=gamma12doublecosetwordlist;
// 
// xi12:=VecSpaceL![0,z-1,1];
// 
// nubetalist:=[[0,4],[0,8],[1,4],[-1,4]];
// Mlist:=["M_c","M_{-c}","b(M_c)","b^-1(M_c)"];
// 
// genlist:=[MatsL | ID3,A1,A1I,A2,A2I,A3,A3I];
// genletters:=[[],["A1"],["A1I"],["A2"],["A2I"],["A3"],["A3I"]];
// possiblefollowingletters:=[
// [1,2,3,4,5,6,7],
// [2,4,5,6,7],
// [3,4,5,6,7],
// [2,3,4,6,7],
// [2,3,5,6,7],
// [2,3,4,5,6],
// [2,3,4,5,7]];
// 
// kalphalist:=[MatsL | kppp,kppm,kpmp,kpmm,kmpp,kmpm,kmmp,kmmm];
// alphalist:=["ppp","ppm","pmp","pmm","mpp","mpm","mmp","mmm"];
// for tidx in [1 .. 72] do
//  print "Doing the case tidx=",tidx;
//  for idx in [1 .. 4] do
//   //print "Case",idx;
//   nu:=nubetalist[idx][1];
//   beta:=nubetalist[idx][2];
//   firstlist:=[MatsL | kalphalist[beta]^-1*B^(-nu)*ID3,kalphalist[beta]^-1*B^(-nu)*A1,kalphalist[beta]^-1*B^(-nu)*A1I,kalphalist[beta]^-1*B^(-nu)*A2,
//                       kalphalist[beta]^-1*B^(-nu)*A2I,kalphalist[beta]^-1*B^(-nu)*A3,kalphalist[beta]^-1*B^(-nu)*A3I];
// 
//   lastlist:=[VecSpaceL | Tlist[tidx]*xi12,A1*Tlist[tidx]*xi12,A1I*Tlist[tidx]*xi12,A2*Tlist[tidx]*xi12,A2I*Tlist[tidx]*xi12,A3*Tlist[tidx]*xi12,A3I*Tlist[tidx]*xi12];
//   for i1 in [1 .. 7] do
//    for i2 in possiblefollowingletters[i1] do
//     tmpmat1:=firstlist[i1]*genlist[i2];
//     for i3 in possiblefollowingletters[i2] do
//      tmpmat2:=tmpmat1*genlist[i3];
//      for i4 in possiblefollowingletters[i3] do
//       tmpmat3:=tmpmat2*genlist[i4];
//       for i5 in possiblefollowingletters[i4] do
//        tmpmat4:=tmpmat3*genlist[i5];
//        for i6 in possiblefollowingletters[i5] do
//         tmpmat5:=tmpmat4*genlist[i6];
//         for i7 in possiblefollowingletters[i6] do
//          tmpmat6:=tmpmat5*genlist[i7];
//          for i8 in possiblefollowingletters[i7] do
//           tmpmat7:=tmpmat6*genlist[i8];
//           for i9 in possiblefollowingletters[i8] do
//            MM:=tmpmat7*lastlist[i9];
//            if (MM[1][1] eq c*MM[2][1]) then
//  print genletters[i1] cat genletters[i2] cat genletters[i3] cat genletters[i4] cat genletters[i5] cat genletters[i6] cat genletters[i7] cat genletters[i8] cat genletters[i9] cat ["hello"] cat Twordlist[tidx],"*xi12 is in ",Mlist[idx];
// //            break i1;
//             break idx;
//            end if;
//           end for;
//          end for;
//         end for;
//        end for;
//       end for;
//      end for;
//     end for;
//    end for;
//   end for;
//  end for;
// end for;


// End of calculations for Proposition 8.
// **************************************

// Calculations for the table in Section 2.8.
// ******************************************
// Calculations with the abelianization map $f:\Pi\to\Z^2$:
// We first set up the space $\Z^2$ of integer pairs $(m,n)$:

Z2:=RMatrixSpace(IntegerRing(),1,2);

// Here are the images in $\Z^2$ of the generators $a_1$, $a_2$ and $a_3$ of $\Pi$:
f1:=Z2![1,3];
f2:=Z2![-2,1];
f3:=Z2![-1,-1];
f1I:=-f1;
f2I:=-f2;
f3I:=-f3;
// 
// To calculate $f(\pi)$ for some specific elements $\pi$, it 
// is useful to first calculate $f(j^4a_ij^{-4})$ and $f(j^{-4}a_ij^4)$,
// as we now do:
// // Recall that the following are zero:
// 
// // J^4*A1*J^8 - z^3*A3*A2I^3*A3^3*A1;
// // J^4*A2*J^8 - (1/z)*A3I;
// // J^4*A3*J^8 - (1/z)*A1I*A2I*A1*A2^2*A1I*A2I*A1*A3I*A1I*A2*A1;
// // 
// // J^4*A1I*J^8 - (1/z^3)*(A1I*A3I*A3I*A3I*A2*A2*A2*A3I);
// // J^4*A2I*J^8 - (z)*A3;
// // J^4*A3I*J^8 - (z)*(A1I*A2I*A1*A3*A1I*A2*A1*A2I*A2I*A1I*A2*A1);
// // 
// // J^8*A1*J^4 - (1/z^4)*A1I*A3I*A3I*A3I*A2*A2*A2*A3I*A1I;
// // J^8*A2*J^4 - A1I*A2I*A1*A3*A1I*A2*A1*A2I*A2I*A1I*A2*A1;
// // J^8*A3*J^4 - (1/z)*A2I;
// //
// // J^8*A1I*J^4 - (z^4)*A1*A3*A2I*A2I*A2I*A3*A3*A3*A1;
// // J^8*A2I*J^4 - A1I*A2I*A1*A2*A2*A1I*A2I*A1*A3I*A1I*A2*A1;
// // J^8*A3I*J^4 - (z)*A2;

// We'll write Bi for J^4*Ai*J^(-4) and Ci for J^(-4)*Ai*J^4:
//
// B1:=z^3*A3*A2I^3*A3^3*A1;
// B2:=(1/z)*A3I;
// B3:=(1/z)*A1I*A2I*A1*A2^2*A1I*A2I*A1*A3I*A1I*A2*A1;
// 
// B1I:=(1/z^3)*(A1I*A3I*A3I*A3I*A2*A2*A2*A3I);
// B2I:=(z)*A3;
// B3I:=(z)*(A1I*A2I*A1*A3*A1I*A2*A1*A2I*A2I*A1I*A2*A1);
// 
// C1:=(1/z^4)*A1I*A3I*A3I*A3I*A2*A2*A2*A3I*A1I;
// C2:=A1I*A2I*A1*A3*A1I*A2*A1*A2I*A2I*A1I*A2*A1;
// C3:=(1/z)*A2I;
// 
// C1I:=(z^4)*A1*A3*A2I*A2I*A2I*A3*A3*A3*A1;
// C2I:=A1I*A2I*A1*A2*A2*A1I*A2I*A1*A3I*A1I*A2*A1;
// C3I:=(z)*A2;

// Here are the images of $j^4a_ij^{-4}$ under $f$:
j4f1:=f3+f2I+f2I+f2I+f3+f3+f3+f1;
j4f2:=f3I;
j4f3:=f1I+f2I+f1+f2+f2+f1I+f2I+f1+f3I+f1I+f2+f1;

j4f1I:=-j4f1;
j4f2I:=-j4f2;
j4f3I:=-j4f3;

// and here are the images of $j^8a_ij^{-8}$ under $f$:
j8f1:=f1I+f3I+f3I+f3I+f2+f2+f2+f3I+f1I;
j8f2:=f1I+f2I+f1+f3+f1I+f2+f1+f2I+f2I+f1I+f2+f1;
j8f3:=f2I;

j8f1I:=-j8f1;
j8f2I:=-j8f2;
j8f3I:=-j8f3;

// Calculations for $\Pi_0$:
// *************************
// Here are the images under the abelianization map f of the generators of $\Pi_0$:

fg1:=f3I+f3I+f3I+f1I+f2+f1;                            // [1,4]
fg3:=f2+f1I+f1I+f3I+f3I+f3I+f1I;                       // [-2,-5]
fg5:=j4f2+j4f1+f2I+f3+f3+f3+f1+f1;                     // [5,-1]
fg7:=j4f1I+j4f2I+j8f2+j8f1;                            // [-7,2]

fg2:=j4f3I+j4f3I+j4f3I+j4f1I+j4f2+j4f1;                // [4,-5]
fg4:=j4f2+j4f1I+j4f1I+j4f3I+j4f3I+j4f3I+j4f1I;         // [-5,7]
fg6:=j8f2+j8f1+j4f2I+j4f3+j4f3+j4f3+j4f1+j4f1;         // [-1,-4]
fg8:=j8f1I+j8f2I+f2+f1;                                // [2,5]

fg1I:=-fg1;
fg2I:=-fg2;
fg3I:=-fg3;
fg4I:=-fg4;
fg5I:=-fg5;
fg6I:=-fg6;
fg7I:=-fg7;
fg8I:=-fg8;

// Recall that the generators $u_i$ and $v_i$ we found satisfying the standard
// surface relation are conjugates of Di's and Ei's, respectively, where

D1:=g1*g2*g3*g4*g5*g6*g7;
D2:=g1*g2*g3*g4;
D3:=g1;
D4:=g3I;

E1:=g8*g1I*g3I*g5I;    
E2:=g5*g6*g2I; 
E3:=g2*g3*g6I;
E4:=g6; 

// Explicitly:
U1:=D1;                            V1:=E1;
U2:=E1*D2*E1^-1;                   V2:=E1*E2*E1^-1;
U3:=E1*E2*D3*E2^-1*E1^-1;          V3:=E1*E2*E3*E2^-1*E1^-1;
U4:=E1*E2*E3*D4*E3^-1*E2^-1*E1^-1; V4:=E1*E2*E3*E4*E3^-1*E2^-1*E1^-1;

// So the following are the images under $f$ of the $D_i$ and $E_i$ for 
// $\Pi_0$, which are the same as the images under $f$ of the $u_i$ and $v_i$, respectively 

fD1:=fg1+fg2+fg3+fg4+fg5+fg6+fg7; // = [-5,-2]
fD2:=fg1+fg2+fg3+fg4;             // = [-2,1]
fD3:=fg1;                         // = [1,4]
fD4:=fg3I;                        // = [2,5]

fE1:=fg8+fg1I+fg3I+fg5I;          // = [-2,7]
fE2:=fg5+fg6+fg2I;                // = [0,0]
fE3:=fg2+fg3+fg6I;                // = [3,-6]
fE4:=fg6;                         // = [-1,-4]

// These are the numbers in the first row of the table in Section 2.8.


// // Now we repeat for $\Pi_1$ and $\Pi_\infty$:
// // The following are zero:

kplus1*g1*kplus1^-1 - (z^2)*A2I*A2I*A2I*A3*A3*A3*A1*A2*A1*A3*A1*A2;
kplus1*g3*kplus1^-1 - (z^3)*J^4*(A2I*A2I*A1I*A3I*A1*A2*A2*A1I*A2I*A1)*J^8;
kplus1*g5*kplus1^-1 - (z^5)*J^8*(A1I*A3I*A1*A2*A2*A1I*A2I*A1*A3*A3)*J^4;
kplus1*g7*kplus1^-1 - (-z)*J^4*(A1I*A1I*A3I*A3I*A3I*A1I*A3I)*J^8;

// The following are zero:
kinfty*g1*kinfty^-1 - (1/z^4)*J^4*A1I*A3I*A3I*A1I*J^8*A1I*A2I;
kinfty*g3*kinfty^-1 - (1/z^2)*J^8*A3*A1*A2*A1I*A2I*J^4;
kinfty*g5*kinfty^-1 - (1/z)*J^8*A2I*A3I*J^4;
kinfty*g7*kinfty^-1 - (1/z^2)*J^4*A1*A3*A1I*A3I*A3I*J^8;

// // So we can get the images under $f$ of the generators $g_i'$ and $g_i''$ of $\Pi_1$ and $\Pi_\infty$:
// // The images of g_1', g_3', g_5' and g_7' are
fg1d:=f2I+f2I+f2I+f3+f3+f3+f1+f2+f1+f3+f1+f2;
fg3d:=j4f2I+j4f2I+j4f1I+j4f3I+j4f1+j4f2+j4f2+j4f1I+j4f2I+j4f1;
fg5d:=j8f1I+j8f3I+j8f1+j8f2+j8f2+j8f1I+j8f2I+j8f1+j8f3+j8f3;
fg7d:=j4f1I+j4f1I+j4f3I+j4f3I+j4f3I+j4f1I+j4f3I;
// 
// // Recall that the following are zero:
// // kplus1*g2*kplus1^-1 - J^4*(kplus1*g1*kplus1^-1)*J^8;
// // kplus1*g4*kplus1^-1 - J^4*(kplus1*g3*kplus1^-1)*J^8;
// // kplus1*g6*kplus1^-1 - J^4*(kplus1*g5*kplus1^-1)*J^8;
// // kplus1*g8*kplus1^-1 - J^4*(kplus1*g7*kplus1^-1)*J^8;
// // So the images of g_2', g_4', g_6' and g_8' are
fg2d:=j4f2I+j4f2I+j4f2I+j4f3+j4f3+j4f3+j4f1+j4f2+j4f1+j4f3+j4f1+j4f2;
fg4d:=j8f2I+j8f2I+j8f1I+j8f3I+j8f1+j8f2+j8f2+j8f1I+j8f2I+j8f1;
fg6d:=f1I+f3I+f1+f2+f2+f1I+f2I+f1+f3+f3;
fg8d:=j8f1I+j8f1I+j8f3I+j8f3I+j8f3I+j8f1I+j8f3I;
// 
// // The following are zero:
// kinfty*g1*kinfty^-1 - (1/z^4)*J^4*A1I*A3I*A3I*A1I*J^8*A1I*A2I;
// kinfty*g3*kinfty^-1 - (1/z^2)*J^8*A3*A1*A2*A1I*A2I*J^4;
// kinfty*g5*kinfty^-1 - (1/z)*J^8*A2I*A3I*J^4;
// kinfty*g7*kinfty^-1 - (1/z^2)*J^4*A1*A3*A1I*A3I*A3I*J^8;
// // The images of g_1'', g_3'', g_5'' and g_7'' are
fg1dd:=j4f1I+j4f3I+j4f3I+j4f1I+f1I+f2I;
fg3dd:=j8f3+j8f1+j8f2+j8f1I+j8f2I;
fg5dd:=j8f2I+j8f3I;
fg7dd:=j4f1+j4f3+j4f1I+j4f3I+j4f3I;
// 
// // Recall that the following are zero:
// // kinfty*g2*kinfty^-1 - J^4*(kinfty*g1*kinfty^-1)*J^8;
// // kinfty*g4*kinfty^-1 - J^4*(kinfty*g3*kinfty^-1)*J^8;
// // kinfty*g6*kinfty^-1 - J^4*(kinfty*g5*kinfty^-1)*J^8;
// // kinfty*g8*kinfty^-1 - J^4*(kinfty*g7*kinfty^-1)*J^8;
// // So we can set
fg2dd:=j8f1I+j8f3I+j8f3I+j8f1I+j4f1I+j4f2I;
fg4dd:=f3+f1+f2+f1I+f2I;
fg6dd:=f2I+f3I;
fg8dd:=j8f1+j8f3+j8f1I+j8f3I+j8f3I;

fg1dI:=-fg1d;
fg2dI:=-fg2d;
fg3dI:=-fg3d;
fg4dI:=-fg4d;
fg5dI:=-fg5d;
fg6dI:=-fg6d;
fg7dI:=-fg7d;
fg8dI:=-fg8d;

fg1ddI:=-fg1dd;
fg2ddI:=-fg2dd;
fg3ddI:=-fg3dd;
fg4ddI:=-fg4dd;
fg5ddI:=-fg5dd;
fg6ddI:=-fg6dd;
fg7ddI:=-fg7dd;
fg8ddI:=-fg8dd;

// So here are the images under $f$ of the $D_i$'s and $E_i$'s of $\Pi_\infty$,
// where are the same as the images under $f$ of the canonical generators $u_i$ and $v_i$.
fD1dd:=fg1dd+fg2dd+fg3dd+fg4dd+fg5dd+fg6dd+fg7dd; // =[-1,2]
fD2dd:=fg1dd+fg2dd+fg3dd+fg4dd;                   // =[-2,1]
fD3dd:=fg1dd;                                     // =[-3,0]
fD4dd:=fg3ddI;                                    // =[-2,1]

fE1dd:=fg8dd+fg1ddI+fg3ddI+fg5ddI;                // =[2,-1]
fE2dd:=fg5dd+fg6dd+fg2ddI;                        // =[0,0]
fE3dd:=fg2dd+fg3dd+fg6ddI;                        // =[-1,2]
fE4dd:=fg6dd;                                     // =[3,0]

// These are the entries in the second row of the table of Section 2.8.

// // So here are the images under $f$ of the $D_i$'s and $E_i$'s of $\Pi_1$,
// // where are the same as the images under $f$ of the canonical generators $u_i$ and $v_i$.
fD1d:=fg1d+fg2d+fg3d+fg4d+fg5d+fg6d+fg7d; // =[-3,0]
fD2d:=fg1d+fg2d+fg3d+fg4d;                // =[2,-1]
fD3d:=fg1d;                               // =[1,4]
fD4d:=fg3dI;                              // =[0,3]

fE1d:=fg8d+fg1dI+fg3dI+fg5dI;             // =[0,3]
fE2d:=fg5d+fg6d+fg2dI;                    // =[-4,2]
fE3d:=fg2d+fg3d+fg6dI;                    // =[7,-8]
fE4d:=fg6d;                               // =[-3,0]
// We haven't included these entries in the table of Section 2.8.

// The case $M=b(M_c)$:
// ********************
// Here are the generators of $\Pi_M$ for $M=b(M_c)$:
P1:=(1/z)*A2*A2*A2*A1I*A3I*J^8*A2I*A2I*A1I*J^4;
P2:=A3*A3*A3*A1*A3*A3*A2*A1*J^4*A3I*J^8*A3I*A3I*A1I*A3I*A3I*A3I;
P3:=z^4*J^8*A1I*A3I*A3I*A3I*A2*A2*J^4*A3I*A3I*A1I*A3I*A3I*A3I;
P4:=(1/z^2)*J^8*A2*A1*A2I*A2I*A1I*J^4*A3*A3*A3*A1*A1*A2I;
P5:=(1/z^4)*A3*A3*A3*A1*A3*A3*J^4*A1I*J^8*A3*A3*A1*A2I*A2I*A2I;
P6:=z^4*A3*A3*A3*A1*A2*A1*A3*A2I*A2I*A2I;
P7:=(1/z)*A3*A3*A3*A1*J^8*A1*A2I*A2I*A1I*A3*A3*J^4;
P8:=(1/z^2)*J^4*A3I*A3I*J^8*A2*A1*A2*A1*A2I*A2I;

P1I:=P1^-1;
P2I:=P2^-1;
P3I:=P3^-1;
P4I:=P4^-1;
P5I:=P5^-1;
P6I:=P6^-1;
P7I:=P7^-1;
P8I:=P8^-1;

print "Checking the relation satisfied by the Pi's:";
P5I*P2I*P5*P1*P3*P8I*P4*P1I*P7I*P6I*P7*P2*P3I*P8*P4I*P6;

// Here are the images under $f$ of the generators of $\Pi_M$ for $M=b(M_c)$:
fP1:=f2+f2+f2+f1I+f3I+j8f2I+j8f2I+j8f1I;                                         // [-4,4]
fP2:=f3+f3+f3+f1+f3+f3+f2+f1+j4f3I+f3I+f3I+f1I+f3I+f3I+f3I;                      // [0,2]
fP3:=j8f1I+j8f3I+j8f3I+j8f3I+j8f2+j8f2+f3I+f3I+f1I+f3I+f3I+f3I;                  // [4,0]
fP4:=j8f2+j8f1+j8f2I+j8f2I+j8f1I+f3+f3+f3+f1+f1+f2I;                             // [0,4]
fP5:=f3+f3+f3+f1+f3+f3+j4f1I+f3+f3+f1+f2I+f2I+f2I;                               // [-2,0]
fP6:=f3+f3+f3+f1+f2+f1+f3+f2I+f2I+f2I;                                           // [2,0]
fP7:=f3+f3+f3+f1+j8f1+j8f2I+j8f2I+j8f1I+j8f3+j8f3;                               // [0,2]
fP8:=j4f3I+j4f3I+f2+f1+f2+f1+f2I+f2I;                                            // [4,2]

fP1I:=-fP1;
fP2I:=-fP2;
fP3I:=-fP3;
fP4I:=-fP4;
fP5I:=-fP5;
fP6I:=-fP6;
fP7I:=-fP7;
fP8I:=-fP8;

// Recall that the generators $u_i$ and $v_i$ we found satisfying the standard
// surface relation are conjugates of Di's and Ei's, respectively, where

D1:=P5I*P2I*P5*P1*P3*P8I*P4*P1I*P7I;
D2:=P5I*P2I*P5*P1*P3*P8I;
D3:=P5I*P2I*P5*P1;
D4:=P5I;

E1:=P6I; 
E2:=P4*P1I*P2*P3I; 
E3:=P3;          
E4:=P2I;

// The following are the images under $f$ of the Di's and Ei's
// used to form the canonical generators u_i and v_i of $\Pi_M$ for $M=b(M_c)$.
// These are the same as the images of the the u_i and v_i, respectively:

fD1:=fP5I+fP2I+fP5+fP1+fP3+fP8I+fP4+fP1I+fP7I; // [0,-2]
fD2:=fP5I+fP2I+fP5+fP1+fP3+fP8I;               // [-4,0]
fD3:=fP5I+fP2I+fP5+fP1;                        // [-4,2]
fD4:=fP5I;                                     // [2,0]

fE1:=fP6I;                                     // [-2,0]
fE2:=fP4+fP1I+fP2+fP3I;                        // [0,2]
fE3:=fP3;                                      // [4,0]
fE4:=fP2I;                                     // [0,-2]

// These are the entries in the third row of the table in Section 2.8.


// The case $M=b^{-1}(M_c)$:
// *************************

// Here are the generators of $\Pi_M$ for $M=b^{-1}(M_c)$:
M1:=z^2*J^8*A1*A3I*A1I*A2I*A2I*J^8*A2*A1*J^8;
M2:=(1/z^3)*J^8*A1*A3*A3*A3*J^4;
M3:=(1/z^5)*J^4*A2I*A2I*A1I*A3I*J^8*A1*A3I*A3I*A1I*A2I*A2I;
M4:=(-1)*J^8*A1*A3I*A1I*A2*A1I*A2I*J^4*A2I*A2I*A3*A3*A3*A1;
M5:=z^4*J^4*A1I*A1I*A3I*A3I*A3I*J^4*A1I*A2I*J^4*A2I*A3*A3*A3*A1;
M6:=(1/z^4)*J^4*A1I*A2I*A1*A3*A3*A3*A1*A2I*J^4*A1I*A2I*A2I*A3*A3*A3*J^4;
M7:=z^2*A3*A1*A2*J^4*A2*A1*J^8;
M8:=(-1)*J^4*A1I*A2I*A1*A2*A2*A1*J^8*A2*A3I*A1I*A2I*A2I;

M1I:=M1^-1;
M2I:=M2^-1;
M3I:=M3^-1;
M4I:=M4^-1;
M5I:=M5^-1;
M6I:=M6^-1;
M7I:=M7^-1;
M8I:=M8^-1;

// Checking the relation satisfied by the $m_i$'s.
// The following is the identity matrix:
M3*M8I*M4*M5*M7I*M2*M3I*M1*M5I*M7*M4I*M1I*M6*M2I*M6I*M8;

// Here are the images under $f$ of the generators of $\Pi_M$ for $M=b^{-1}(M_c)$:
fM1:=j8f1+j8f3I+j8f1I+j8f2I+j8f2I+j4f2+j4f1;                                     // [0,2]
fM2:=j8f1+j8f3+j8f3+j8f3;                                                        // [2,-2]
fM3:=j4f2I+j4f2I+j4f1I+j4f3I+f1+f3I+f3I+f1I+f2I+f2I;                             // [2,0]
fM4:=j8f1+j8f3I+j8f1I+j8f2+j8f1I+j8f2I+f2I+f2I+f3+f3+f3+f1;                      // [4,-2]
fM5:=j4f1I+j4f1I+j4f3I+j4f3I+j4f3I+j8f1I+j8f2I+f2I+f3+f3+f3+f1;                  // [0,2]
fM6:=j4f1I+j4f2I+j4f1+j4f3+j4f3+j4f3+j4f1+j4f2I+j8f1I+j8f2I+j8f2I+j8f3+j8f3+j8f3;// [6,0]
fM7:=f3+f1+f2+j4f2+j4f1;                                                         // [2,0]
fM8:=j4f1I+j4f2I+j4f1+j4f2+j4f2+j4f1+f2+f3I+f1I+f2I+f2I;                         // [6,-6]

fM1I:=-fM1;
fM2I:=-fM2;
fM3I:=-fM3;
fM4I:=-fM4;
fM5I:=-fM5;
fM6I:=-fM6;
fM7I:=-fM7;
fM8I:=-fM8;

// Recall that the generators $u_i$ and $v_i$ for $\Pi_M$ for $M=b^{-1}(M_c)$
// were found satisfying the standard surface relation. They are conjugates of 
// Di's and Ei's, respectively, where

D1:=M3;
D2:=M1*M5I*M7*M4I*M1I*M6;
D3:=M1*M5I;
D4:=M1;

E1:=M8I*M4*M5*M7I*M2;
E2:=M2I;
E3:=M7*M4I*M1I*M4;
E4:=M4I;

// Here are the images under $f$ of the $D_i$ and $E_i$
// of $\Pi_M$, where $M=b^{-1}(M_c)$.

fD1:=fM3;                         // [2,0]
fD2:=fM1+fM5I+fM7+fM4I+fM1I+fM6;  // [4,0]
fD3:=fM1+fM5I;                    // [0,0]
fD4:=fM1;                         // [0,2]

fE1:=fM8I+fM4+fM5+fM7I+fM2;       // [-2,4]
fE2:=fM2I;                        // [-2,2]
fE3:=fM7+fM4I+fM1I+fM4;           // [2,-2]
fE4:=fM4I;                        // [-4,2]

// These values were omitted from the table in Section 2.8.

// Calculations with the Abelianization map $f:\Pi\to\Z^2$
// related to $M_c$ and $M_{-c}$.

// Recall that the following are J^4*Ai*J^(-4)'s:
B1:=z^3*A3*A2I^3*A3^3*A1;
B2:=(1/z)*A3I;
B3:=(1/z)*A1I*A2I*A1*A2^2*A1I*A2I*A1*A3I*A1I*A2*A1;

B1I:=(1/z^3)*(A1I*A3I*A3I*A3I*A2*A2*A2*A3I);
B2I:=(z)*A3;
B3I:=(z)*(A1I*A2I*A1*A3*A1I*A2*A1*A2I*A2I*A1I*A2*A1);

// Recall that the following are J^(-4)*Ai*J^4's:
C1:=(1/z^4)*A1I*A3I*A3I*A3I*A2*A2*A2*A3I*A1I;
C2:=A1I*A2I*A1*A3*A1I*A2*A1*A2I*A2I*A1I*A2*A1;
C3:=(1/z)*A2I;

C1I:=(z^4)*A1*A3*A2I*A2I*A2I*A3*A3*A3*A1;
C2I:=A1I*A2I*A1*A2*A2*A1I*A2I*A1*A3I*A1I*A2*A1;
C3I:=(z)*A2;

// Recall that the 20 generators $g_i$ found for $\Pi_c$
// we described by giving $g_i$ for $i=1,3,5,7,9,12,15,17,19$,
// and then using conjugation by $j^4$.

// The following are zero:
Gc1 - (C1I*C2*C1*C3*C1I*A2*A1);
Gc3 - z*(B2*B1*B2I*B2I*B1I*B3*C3*C3*C3);
Gc5 - z^4*(C1I*A2*A1*B3*B2I*B1*B3*B1I);
Gc7 - (1/z^5)*(C2*C1*A3I*B2*B1I*B2I*B3I*B3I*B3I);
Gc9 - z^4*(C1I*C2I*C2I*C1I*C3I*B1I*B2I);
Gc12 - (-1)*(A2I*A1*A3*A1I*A3I*B3*B1*B2*B2*B1I*B2I);
Gc15 - (z)*(B1*C2*C3*C1I);
Gc17 - (1/z^2)*(C1I*C1I*C2I*A3*A1*A2*A1);
Gc19 - (1/z)*(A2I*A1*A3*A1I*A3I*A3I*B1*B2*C1I*C2I);
print "Finished expressing the Gc_i in terms of the Ai,Bi and Ci,";
print "where Bi=J^4*Ai*J^(-4) and Ci = J^8*Ai*J^(-8):";

// Here are the images of $j^4a_ij^{-4}$ under $f$:
j4f1:=f3+f2I+f2I+f2I+f3+f3+f3+f1;
j4f2:=f3I;
j4f3:=f1I+f2I+f1+f2+f2+f1I+f2I+f1+f3I+f1I+f2+f1;

j4f1I:=-j4f1;
j4f2I:=-j4f2;
j4f3I:=-j4f3;

// and here are the images of $j^8a_ij^{-8}$ under $f$:
j8f1:=f1I+f3I+f3I+f3I+f2+f2+f2+f3I+f1I;
j8f2:=f1I+f2I+f1+f3+f1I+f2+f1+f2I+f2I+f1I+f2+f1;
j8f3:=f2I;

j8f1I:=-j8f1;
j8f2I:=-j8f2;
j8f3I:=-j8f3;

// So here are the images under $f$ of the 9 generators $g_i$, $i=1,3,5,7,9,12,15,17,19$:
fgc1:=j8f1I+j8f2+j8f1+j8f3+j8f1I+f2+f1;
fgc3:=j4f2+j4f1+j4f2I+j4f2I+j4f1I+j4f3+j8f3+j8f3+j8f3;
fgc5:=j8f1I+f2+f1+j4f3+j4f2I+j4f1+j4f3+j4f1I;
fgc7:=j8f2+j8f1+f3I+j4f2+j4f1I+j4f2I+j4f3I+j4f3I+j4f3I;
fgc9:=j8f1I+j8f2I+j8f2I+j8f1I+j8f3I+j4f1I+j4f2I;
fgc12:=f2I+f1+f3+f1I+f3I+j4f3+j4f1+j4f2+j4f2+j4f1I+j4f2I;
fgc15:=j4f1+j8f2+j8f3+j8f1I;
fgc17:=j8f1I+j8f1I+j8f2I+f3+f1+f2+f1;
fgc19:=f2I+f1+f3+f1I+f3I+f3I+j4f1+j4f2+j8f1I+j8f2I;

// From these, we get the images under $f$ of the other 11 generators $g_i$:

fgc2:=f1I+f2+f1+f3+f1I+j4f2+j4f1;
fgc4:=j8f2+j8f1+j8f2I+j8f2I+j8f1I+j8f3+f3+f3+f3;
fgc6:=f1I+j4f2+j4f1+j8f3+j8f2I+j8f1+j8f3+j8f1I;
fgc8:=f2+f1+j4f3I+j8f2+j8f1I+j8f2I+j8f3I+j8f3I+j8f3I;
fgc10:=f1I+f2I+f2I+f1I+f3I+j8f1I+j8f2I;
fgc11:=j4f1I+j4f2I+j4f2I+j4f1I+j4f3I+f1I+f2I;
fgc13:=j4f2I+j4f1+j4f3+j4f1I+j4f3I+j8f3+j8f1+j8f2+j8f2+j8f1I+j8f2I;
fgc14:=j8f2I+j8f1+j8f3+j8f1I+j8f3I+f3+f1+f2+f2+f1I+f2I;
fgc16:=j8f1+f2+f3+f1I;
fgc18:=f1I+f1I+f2I+j4f3+j4f1+j4f2+j4f1;
fgc20:=j4f2I+j4f1+j4f3+j4f1I+j4f3I+j4f3I+j8f1+j8f2+f1I+f2I;

fgc1I:=-fgc1;
fgc2I:=-fgc2;
fgc3I:=-fgc3;
fgc4I:=-fgc4;
fgc5I:=-fgc5;
fgc6I:=-fgc6;
fgc7I:=-fgc7;
fgc8I:=-fgc8;
fgc9I:=-fgc9;
fgc10I:=-fgc10;
fgc11I:=-fgc11;
fgc12I:=-fgc12;
fgc13I:=-fgc13;
fgc14I:=-fgc14;
fgc15I:=-fgc15;
fgc16I:=-fgc16;
fgc17I:=-fgc17;
fgc18I:=-fgc18;
fgc19I:=-fgc19;
fgc20I:=-fgc20;

print "Finished calculating the images under f of the";
print "generators g_1,...,g_(20) of Pi_c.";

// Checking Formula (3) at the end of Section 1.5:
// ***********************************************
Mats2Z:=MatrixRing(IntegerRing(),2);

MM:=Mats2Z![0,-1,1,-1];

// The following are zero:

j4f1-f1*MM;
j4f2-f2*MM;
j4f3-f3*MM;

j8f1-j4f1*MM;
j8f2-j4f2*MM;
j8f3-j4f3*MM;

fgc2 - fgc1*MM;
fgc4 - fgc3*MM;
fgc6 - fgc5*MM;
fgc8 - fgc7*MM;
fgc10 - fgc9*MM;
fgc11 - fgc10*MM;
fgc13 - fgc12*MM;
fgc14 - fgc13*MM;
fgc16 - fgc15*MM;
fgc18 - fgc17*MM;
fgc20 - fgc19*MM;

// The following is the identity matrix:

Gc4*Gc14I*Gc2I*Gc17I*Gc9*Gc19*Gc20*Gc14*Gc7I*Gc10I*Gc5I* Gc16I*Gc3I* Gc12I*Gc1*Gc2*Gc18I*Gc10*Gc19I*Gc12
 *Gc8I*Gc11I*Gc6I*Gc15*Gc16*Gc4I*Gc13I*Gc1I*Gc17*Gc18*Gc11*Gc20I*Gc13*Gc7*Gc8*Gc9I*Gc5*Gc6*Gc15I*Gc3;
print "word equals ID3";

// Recall that generators $u_i$ and $v_i$ for $\Pi_c$ were found satisfying the standard
// surface group relation. They are conjugates in $\Pi_c$ of Di's and Ei's, i=1,...,10,
// where 
D1:=Gc4*Gc14I*Gc2I*Gc17I*Gc9*Gc19*Gc20*Gc14*Gc7I*Gc10I*Gc5I*Gc16I;
D2:=Gc4*Gc14I*Gc2I*Gc17I*Gc9*Gc19*Gc20*Gc14*Gc7I*Gc10I*Gc5I*Gc4I*Gc13I*Gc1I*Gc17*Gc18*Gc11*Gc20I*Gc13*Gc7*Gc8*Gc9I*Gc5*Gc6;
D3:=Gc4*Gc14I*Gc2I*Gc17I*Gc9*Gc19*Gc20*Gc14*Gc7I*Gc10I*Gc5I*Gc4I*Gc13I*Gc1I*Gc17*Gc18;
D4:=Gc4*Gc14I;
D5:=Gc4*Gc7I*Gc10I*Gc5I*Gc4I*Gc13I*Gc1I*Gc17*Gc10*Gc19I*Gc12*Gc8I;
D6:=Gc4;
D7:=Gc13I*Gc1I*Gc17*Gc10*Gc19I*Gc12*Gc9I;
D8:=Gc13I*Gc1I;
D9:=Gc13I*Gc10*Gc19I;
D10:=Gc13I;

E1:=Gc3I*Gc12I*Gc1*Gc2*Gc18I*Gc10*Gc19I*Gc12*Gc8I*Gc11I*Gc6I*Gc15;
E2:=Gc15I*Gc12I*Gc1*Gc2*Gc18I*Gc10*Gc19I*Gc12*Gc8I*Gc11I;
E3:=Gc11*Gc20I*Gc13*Gc7*Gc8*Gc9I*Gc5*Gc12I*Gc1*Gc2;
E4:=Gc2I*Gc17I*Gc9*Gc19*Gc20;
E5:=Gc20I*Gc13*Gc7;
E6:=Gc7I*Gc10I*Gc5I;
E7:=Gc5*Gc12I*Gc1*Gc17I;
E8:=Gc17*Gc10*Gc19I*Gc12*Gc19*Gc13*Gc10I*Gc12I;
E9:=Gc12;
E10:=Gc10;
// The images under $f$ of the $u_i$'s and $v_i$'s are the same as the images under $f$ of the
// Di's and Ei's, which we now calculate:

// Calculating the images of the Di's and Ei's under the abelianization map:

fD1:=fgc4+fgc14I+fgc2I+fgc17I+fgc9+fgc19+fgc20+fgc14+fgc7I+fgc10I+fgc5I+fgc16I;                         // [4,-2]
fD2:=fgc4+fgc14I+fgc2I+fgc17I+fgc9+fgc19+fgc20+fgc14+fgc7I+fgc10I+fgc5I+fgc4I     
     +fgc13I+fgc1I+fgc17+fgc18+fgc11+fgc20I+fgc13+fgc7+fgc8+fgc9I+fgc5+fgc6;                            // [2,-4]
fD3:=fgc4+fgc14I+fgc2I+fgc17I+fgc9+fgc19+fgc20+fgc14+fgc7I+fgc10I+fgc5I+fgc4I+fgc13I+fgc1I+fgc17+fgc18; // [2,-4]
fD4:=fgc4+fgc14I;                                                                                       // [2,-4]
fD5:=fgc4+fgc7I+fgc10I+fgc5I+fgc4I+fgc13I+fgc1I+fgc17+fgc10+fgc19I+fgc12+fgc8I;                         // [-6,6]

fD6:=fgc4;                                                                                              // [-2,-2]
fD7:=fgc13I+fgc1I+fgc17+fgc10+fgc19I+fgc12+fgc9I;                                                       // [-4,2]
fD8:=fgc13I+fgc1I;                                                                                      // [-8,4]
fD9:=fgc13I+fgc10+fgc19I;                                                                               // [-6,0]
fD10:=fgc13I;                                                                                           // [-2,4]

fE1:=fgc3I+fgc12I+fgc1+fgc2+fgc18I+fgc10+fgc19I+fgc12+fgc8I+fgc11I+fgc6I+fgc15;                         // [4,-2]
fE2:=fgc15I+fgc12I+fgc1+fgc2+fgc18I+fgc10+fgc19I+fgc12+fgc8I+fgc11I;                                    // [-6,6]
fE3:=fgc11+fgc20I+fgc13+fgc7+fgc8+fgc9I+fgc5+fgc12I+fgc1+fgc2;                                          // [-2,-2]
fE4:=fgc2I+fgc17I+fgc9+fgc19+fgc20;                                                                     // [2,-4]
fE5:=fgc20I+fgc13+fgc7;                                                                                 // [2,2]

fE6:=fgc7I+fgc10I+fgc5I;                                                                                // [-4,2]
fE7:=fgc5+fgc12I+fgc1+fgc17I;                                                                           // [-2,-2]
fE8:=fgc17+fgc10+fgc19I+fgc12+fgc19+fgc13+fgc10I+fgc12I;                                                // [8,2]
fE9:=fgc12;                                                                                             // [2,2]
fE10:=fgc10;                                                                                            // [6,-6]

// The above values, and those calculated next, are the values remarked on right after the table of Section 2.8. 

// Here are generators for $\bar\Gamma_{-c}$:
// The following are zero:
kmmm*Gc1*kmmm^-1 - J^4*A1I*A3I*A3I*A3I*A2*A2*J^4*A2I*A2I*A3*A3*A3*A1*J^4;
kmmm*Gc3*kmmm^-1 - z^2*J^8*A2*A2*A3*A3*J^4;
kmmm*Gc5*kmmm^-1 - (1/z^4)*J^8*A1I*A2I*A1*A2*A2*J^8*A1I*A3I*A3I*A3I*J^8*A2*A1;
kmmm*Gc7*kmmm^-1 - (1/z^3)*J^8*A1*A2*A2*A1I*A2I*J^8*A3I*A1I*A2*A1*A2I*J^8;
kmmm*Gc9*kmmm^-1 - (1/z^2)*J^8*A1I*A2I*A3I*A3I*A1I*J^8*A2I*A2I*A1I*J^8;
kmmm*Gc12*kmmm^-1 - J^4*A1I*A3I*A3I*A3I*A2*A2*J^8*A1I*J^4*A2*A1I*A3*J^8;
kmmm*Gc15*kmmm^-1 - z^4*J^8*A2*A1*A3*A1I*J^4*A1I*A3I*A3I*A3I*A1I*A1I*A3I*A3I*A3I;
kmmm*Gc17*kmmm^-1 - (1/z^2)*J^8*A1I*A2I*J^4*A1I*A3I*A3I*A3I*J^8*A1*A2I*A3*A3*A3*A1*J^4;
kmmm*Gc19*kmmm^-1 - (1/z^2)*J^4*A1*A2*A1I*A1I*A3I*A3I*J^4*A2*A1I*A2I*J^8*A2*A2*A1I*A2I*J^8*A2I;

print "Finished finding words for the generators of Pi_(-c):";
print "The following are zero matrices:";

kmmm*Gc1*kmmm^-1 - B1I*B3I*B3I*B3I*B2*B2*C2I*C2I*C3*C3*C3*C1;
kmmm*Gc3*kmmm^-1 - z^2*C2*C2*C3*C3;
kmmm*Gc5*kmmm^-1 - (1/z^4)*C1I*C2I*C1*C2*C2*B1I*B3I*B3I*B3I*A2*A1;
kmmm*Gc7*kmmm^-1 - (1/z^3)*C1*C2*C2*C1I*C2I*B3I*B1I*B2*B1*B2I;
kmmm*Gc9*kmmm^-1 - (1/z^2)*C1I*C2I*C3I*C3I*C1I*B2I*B2I*B1I;
kmmm*Gc12*kmmm^-1 - B1I*B3I*B3I*B3I*B2*B2*A1I*B2*B1I*B3;
kmmm*Gc15*kmmm^-1 - z^4*C2*C1*C3*C1I*A1I*A3I*A3I*A3I*A1I*A1I*A3I*A3I*A3I;
kmmm*Gc17*kmmm^-1 - (1/z^2)*C1I*C2I*A1I*A3I*A3I*A3I*C1*C2I*C3*C3*C3*C1;
kmmm*Gc19*kmmm^-1 - (1/z^2)*B1*B2*B1I*B1I*B3I*B3I*C2*C1I*C2I*B2*B2*B1I*B2I*A2I;

print "Finished finding expressions in Ai, Bi and Ci for the generators of Pi_(-c):";

fgc1d:=j4f1I+j4f3I+j4f3I+j4f3I+j4f2+j4f2+j8f2I+j8f2I+j8f3+j8f3+j8f3+j8f1;
fgc3d:=j8f2+j8f2+j8f3+j8f3;
fgc5d:=j8f1I+j8f2I+j8f1+j8f2+j8f2+j4f1I+j4f3I+j4f3I+j4f3I+f2+f1;
fgc7d:=j8f1+j8f2+j8f2+j8f1I+j8f2I+j4f3I+j4f1I+j4f2+j4f1+j4f2I;
fgc9d:=j8f1I+j8f2I+j8f3I+j8f3I+j8f1I+j4f2I+j4f2I+j4f1I;
fgc12d:=j4f1I+j4f3I+j4f3I+j4f3I+j4f2+j4f2+f1I+j4f2+j4f1I+j4f3;
fgc15d:=j8f2+j8f1+j8f3+j8f1I+f1I+f3I+f3I+f3I+f1I+f1I+f3I+f3I+f3I;
fgc17d:=j8f1I+j8f2I+f1I+f3I+f3I+f3I+j8f1+j8f2I+j8f3+j8f3+j8f3+j8f1;
fgc19d:=j4f1+j4f2+j4f1I+j4f1I+j4f3I+j4f3I+j8f2+j8f1I+j8f2I+j4f2+j4f2+j4f1I+j4f2I+f2I;

fgc2d:=j8f1I+j8f3I+j8f3I+j8f3I+j8f2+j8f2+f2I+f2I+f3+f3+f3+f1;
fgc4d:=f2+f2+f3+f3;
fgc6d:=f1I+f2I+f1+f2+f2+j8f1I+j8f3I+j8f3I+j8f3I+j4f2+j4f1;
fgc8d:=f1+f2+f2+f1I+f2I+j8f3I+j8f1I+j8f2+j8f1+j8f2I;
fgc10d:=f1I+f2I+f3I+f3I+f1I+j8f2I+j8f2I+j8f1I;
fgc11d:=j4f1I+j4f2I+j4f3I+j4f3I+j4f1I+f2I+f2I+f1I;
fgc13d:=j8f1I+j8f3I+j8f3I+j8f3I+j8f2+j8f2+j4f1I+j8f2+j8f1I+j8f3;
fgc14d:=f1I+f3I+f3I+f3I+f2+f2+j8f1I+f2+f1I+f3;
fgc16d:=f2+f1+f3+f1I+j4f1I+j4f3I+j4f3I+j4f3I+j4f1I+j4f1I+j4f3I+j4f3I+j4f3I;
fgc18d:=f1I+f2I+j4f1I+j4f3I+j4f3I+j4f3I+f1+f2I+f3+f3+f3+f1;
fgc20d:=j8f1+j8f2+j8f1I+j8f1I+j8f3I+j8f3I+f2+f1I+f2I+j8f2+j8f2+j8f1I+j8f2I+j4f2I;

fgc1d:=j4f1I+j4f3I+j4f3I+j4f3I+j4f2+j4f2+j8f2I+j8f2I+j8f3+j8f3+j8f3+j8f1;            // [2,2]
fgc2d:=j8f1I+j8f3I+j8f3I+j8f3I+j8f2+j8f2+f2I+f2I+f3+f3+f3+f1;                        // [2,-4]
fgc3d:=j8f2+j8f2+j8f3+j8f3;                                                          // [6,-6]
fgc4d:=f2+f2+f3+f3;                                                                  // [-6,0]
fgc5d:=j8f1I+j8f2I+j8f1+j8f2+j8f2+j4f1I+j4f3I+j4f3I+j4f3I+f2+f1;                     // [0,0]
fgc6d:=f1I+f2I+f1+f2+f2+j8f1I+j8f3I+j8f3I+j8f3I+j4f2+j4f1;                           // [0,0]
fgc7d:=j8f1+j8f2+j8f2+j8f1I+j8f2I+j4f3I+j4f1I+j4f2+j4f1+j4f2I;                       // [2,-4]
fgc8d:=f1+f2+f2+f1I+f2I+j8f3I+j8f1I+j8f2+j8f1+j8f2I;                                 // [-4,2]
fgc9d:=j8f1I+j8f2I+j8f3I+j8f3I+j8f1I+j4f2I+j4f2I+j4f1I;                              // [-2,4]
fgc10d:=f1I+f2I+f3I+f3I+f1I+j8f2I+j8f2I+j8f1I;                                       // [4,-2]
fgc11d:=j4f1I+j4f2I+j4f3I+j4f3I+j4f1I+f2I+f2I+f1I;                                   // [-2,-2]
fgc12d:=j4f1I+j4f3I+j4f3I+j4f3I+j4f2+j4f2+f1I+j4f2+j4f1I+j4f3;                       // [-2,4]
fgc13d:=j8f1I+j8f3I+j8f3I+j8f3I+j8f2+j8f2+j4f1I+j8f2+j8f1I+j8f3;                     // [4,-2]
fgc14d:=f1I+f3I+f3I+f3I+f2+f2+j8f1I+f2+f1I+f3;                                       // [-2,-2]
fgc15d:=j8f2+j8f1+j8f3+j8f1I+f1I+f3I+f3I+f3I+f1I+f1I+f3I+f3I+f3I;                    // [6,-6]
fgc16d:=f2+f1+f3+f1I+j4f1I+j4f3I+j4f3I+j4f3I+j4f1I+j4f1I+j4f3I+j4f3I+j4f3I;          // [-6,0]
fgc17d:=j8f1I+j8f2I+f1I+f3I+f3I+f3I+j8f1+j8f2I+j8f3+j8f3+j8f3+j8f1;                  // [2,2]
fgc18d:=f1I+f2I+j4f1I+j4f3I+j4f3I+j4f3I+f1+f2I+f3+f3+f3+f1;                          // [2,-4]
fgc19d:=j4f1+j4f2+j4f1I+j4f1I+j4f3I+j4f3I+j8f2+j8f1I+j8f2I+j4f2+j4f2+j4f1I+j4f2I+f2I;// [4,4]
fgc20d:=j8f1+j8f2+j8f1I+j8f1I+j8f3I+j8f3I+f2+f1I+f2I+j8f2+j8f2+j8f1I+j8f2I+j4f2I;    // [4,-8]

fgc1dI:=-fgc1d;
fgc2dI:=-fgc2d;
fgc3dI:=-fgc3d;
fgc4dI:=-fgc4d;
fgc5dI:=-fgc5d;
fgc6dI:=-fgc6d;
fgc7dI:=-fgc7d;
fgc8dI:=-fgc8d;
fgc9dI:=-fgc9d;
fgc10dI:=-fgc10d;
fgc11dI:=-fgc11d;
fgc12dI:=-fgc12d;
fgc13dI:=-fgc13d;
fgc14dI:=-fgc14d;
fgc15dI:=-fgc15d;
fgc16dI:=-fgc16d;
fgc17dI:=-fgc17d;
fgc18dI:=-fgc18d;
fgc19dI:=-fgc19d;
fgc20dI:=-fgc20d;


// Recall that generators $u_i$ and $v_i$ for $\Pi_{-c}$ were found satisfying the standard
// surface group relation. They are conjugates in $\Pi_{-c}$ of Di's and Ei's, i=1,...,10,
// which are obtained from the original generators of $\Pi_{-c}$ in exactly the same way
// as was done for the group $\Pi_c$.

// Calculating the images of the Di's and Ei's under the abelianization map:

fD1d:=fgc4d+fgc14dI+fgc2dI+fgc17dI+fgc9d+fgc19d+fgc20d+fgc14d+fgc7dI+fgc10dI+fgc5dI+fgc16dI;                             // [-4,8]
fD2d:=fgc4d+fgc14dI+fgc2dI+fgc17dI+fgc9d+fgc19d+fgc20d+fgc14d+fgc7dI+fgc10dI+fgc5dI+fgc4dI     
      +fgc13dI+fgc1dI+fgc17d+fgc18d+fgc11d+fgc20dI+fgc13d+fgc7d+fgc8d+fgc9dI+fgc5d+fgc6d;                                // [-8,4]
fD3d:=fgc4d+fgc14dI+fgc2dI+fgc17dI+fgc9d+fgc19d+fgc20d+fgc14d+fgc7dI+fgc10dI+fgc5dI+fgc4dI+fgc13dI+fgc1dI+fgc17d+fgc18d; // [-6,6]
fD4d:=fgc4d+fgc14dI;                                                                                                     // [-4,2]
fD5d:=fgc4d+fgc7dI+fgc10dI+fgc5dI+fgc4dI+fgc13dI+fgc1dI+fgc17d+fgc10d+fgc19dI+fgc12d+fgc8dI;                             // [-8,4]

fD6d:=fgc4d;                                                                                                             // [-6,0]
fD7d:=fgc13dI+fgc1dI+fgc17d+fgc10d+fgc19dI+fgc12d+fgc9dI;                                                                // [-4,-4]
fD8d:=fgc13dI+fgc1dI;                                                                                                    // [-6,0]
fD9d:=fgc13dI+fgc10d+fgc19dI;                                                                                            // [-4,-4]
fD10d:=fgc13dI;                                                                                                          // [-4,2]

fE1d:=fgc3dI+fgc12dI+fgc1d+fgc2d+fgc18dI+fgc10d+fgc19dI+fgc12d+fgc8dI+fgc11dI+fgc6dI+fgc15d;                             // [8,-4]
fE2d:=fgc15dI+fgc12dI+fgc1d+fgc2d+fgc18dI+fgc10d+fgc19dI+fgc12d+fgc8dI+fgc11dI;                                          // [2,2]
fE3d:=fgc11d+fgc20dI+fgc13d+fgc7d+fgc8d+fgc9dI+fgc5d+fgc12dI+fgc1d+fgc2d;                                                // [4,-8]
fE4d:=fgc2dI+fgc17dI+fgc9d+fgc19d+fgc20d;                                                                                // [2,2]
fE5d:=fgc20dI+fgc13d+fgc7d;                                                                                              // [2,2]

fE6d:=fgc7dI+fgc10dI+fgc5dI;                                                                                             // [-6,6]
fE7d:=fgc5d+fgc12dI+fgc1d+fgc17dI;                                                                                       // [2,-4]
fE8d:=fgc17d+fgc10d+fgc19dI+fgc12d+fgc19d+fgc13d+fgc10dI+fgc12dI;                                                        // [6,0]
fE9d:=fgc12d;                                                                                                            // [-2,4]
fE10d:=fgc10d;                                                                                                           // [4,-2]

// These values, and those calculated for $\bar\Gamma_c$ above, are the values remarked on right after the table of Section 2.8. 

// End of calculations for the table in Section 2.8.
// *************************************************

// Calculations for Proposition 11.
// ********************************

// The following elements are in $\Pi$, as shown in the file
// cs-surface-presentations_magma.txt.

elt1:=B^-1*(V*U*J*J*J)*(B*U*V)*(V*U*J*J*J)^-1*B^1*J^-4;
elt2:=B^0*(UI*V*J)*(B*U*V)*(UI*V*J)^-1*B^0*J^-4;
elt3:=B^1*(U*V*V*J*J)*(B*U*V)*(U*V*V*J*J)^-1*B^-1*J^-4;
elt4:=B^-1*(V*V*U*J*J*J)*(B*U*V)*(V*V*U*J*J*J)^-1*B^1*J^-4;
elt5:=B^0*(V*J*J)*(B*U*V)*(V*J*J)^-1*B^0*J^-4;
elt6:=B^1*(V*UI*V)*(B*U*V)*(V*UI*V)^-1*B^-1*J^-4;

// Here we confirm that the six elements are in $\Pi$ by exhibiting
// words for each of them in the generators of $\Pi$.
// The following are zero:
elt1 - z^4*A2*A2*A1*A3*A3*A3;
elt2 - J^8*A1*J^4;
elt3 - z^2*J^8*A1*A2*A2*A2*J^4*A2*A1*A2I*A2I*A1I;
elt4 - (1/z^5)*A3*A3*A3*A1*A1*A3*A3*A3;
elt5 - (1/z)*J^4*A1I*A2I*J^8;
elt6 - z*A2*A1I;

// Here are the three elements $\pi_0$, $\pi_1$ and $\pi_{-1}$
// appearing in Equation (14), in the proof of Proposition 11.

// The following are zero, showing that 
// $b^\mu j^4b^{-\mu}j^{-4}=\pi_\nu\in\Pi$ for $\pi_0=1$,
// $\pi_1=a_2a_1^{-2}a_3^{-3}a_1^{-1}$ and 
// $\pi_{-1}=a_2^2a_1a_3a_1^{-1}$.

pi0:=ID3;
pi1:=(1/z^4)*A2*A1I^2*A3I^3*A1I;
pim1:=A2^2*A1*A3*A1I;
// The following are zero:
B^0*J^4*BI^0*JI^4 - pi0;
B*J^4*BI*JI^4 - pi1;
BI*J^4*B*JI^4 - pim1;

// Here are the images under the abelianization map $f$ of $\pi_0$, $\pi_1$ and $\pi_{-1}$:

fpi0:=Z2!0;
fpi1:=f2+f1I+f1I+f3I+f3I+f3I+f1I;
fpim1:=f2+f2+f1+f3+f1I;

// Here are the images under $f$ of the elements elt1,...,elt6 above.
felt1 := f2+f2+f1+f3+f3+f3;                     // (-6,2)
felt2 := j8f1;                                  // (-4,1)
felt3 := j8f1+j8f2+j8f2+j8f2+f2+f1+f2I+f2I+f1I; // (1,-6)
felt4 := f3+f3+f3+f1+f1+f3+f3+f3;               // (-4,0)
felt5 := j4f1I+j4f2I;                           // (-4,3)
felt6 := f2+f1I;                                // (-3,-2)

// End of calculations for Proposition 11.
// ***************************************