let h^2=7;      % Square root of 2
write "h^2=",h^2;
let z^3=(h*i-1)*z^2/2+(h*i+1)*z/2+1;

write "z^7=",z^7;

write "z+z^2-z^3+z^4-z^5-z^6 = ",z+z^2-z^3+z^4-z^5-z^6;

W:=z+z^6;
write "W^3+W^2-2*W-1 = ",W^3+W^2-2*W-1;
phiW:=W^2-2$
phi2W:=-W^2-W+1$

dt:=(3+h*i)/4$
dti:=(3-h*i)/4$  
write "dt*dti = ", dt*dti; 

ID3:=mat(
(1,0,0),
(0,1,0),
(0,0,1))$

SIG:=mat(
(0,1,0),
(0,0,1),
(dt,0,0))$

% Here is the inverse of SIG:

SIGI:=dti*SIG^2$

T:=(2-4*h/7) + (1-3*h/7)*W  + (-h/7)*W^2$
phiT:=(-h/7)   + (h/7)*W     + (1-2*h/7)*W^2$
phi2T:=(3-9*h/7) + (-1+2*h/7)*W + (-1+3*h/7)*W^2$

TI:=1+(2/7)*h+(-1-(2/7)*h)*W+(-1-(3/7)*h)*W^2$
phiTI:=(-3*h+(3*h+7)*W+h*W^2)/7$
phi2TI:=(-2*(3*h+7)-h*W+(7+2*h)*W^2)/7$

% The following are zero, verifying that phiT/T and phi2T/T 
% are algebraic integers.
sub(x=phiT/T,x^3+(2*h-3)*x^2+(-h+3)*x-1);
sub(x=phi2T/T,x^3+(h-3)*x^2+(-2*h+3)*x-1);

tildeT:=(2+4*h/7) + (1+3*h/7)*W  + (h/7)*W^2$
phitildeT:=(2+4*h/7) + (1+3*h/7)*phiW  + (h/7)*phiW^2$
phi2tildeT:=(2+4*h/7) + (1+3*h/7)*phi2W  + (h/7)*phi2W^2$

% The following are zero, verifying that phitildeT/tildeT and phi2tildeT/tildeT 
% are algebraic integers.
sub(x=phitildeT/tildeT,x^3+(-2*h-3)*x^2+(h+3)*x-1);
sub(x=phi2tildeT/tildeT,x^3+(-h-3)*x^2+(2*h+3)*x-1);

FF:=mat(
(T,   0,    0),
(0,phiT,    0),
(0,   0,phi2T))$

FFI:=mat(
(TI,   0,    0),
( 0,phiTI,    0),
( 0,   0,phi2TI))$

write "FF*FFI = ", FF*FFI; 

ZMAT:=mat(
(z,  0,  0),
(0,z^2,  0),
(0,  0,z^4))$

ZMATI:=mat(
(z^6,  0,  0),
(  0,z^5,  0),
(  0,  0,z^3))$

write "ZMAT*ZMATI = ",ZMAT*ZMATI;

SS:=((c00m+c10m*i)*ID3+(c01m+c11m*i)*ZMAT+(c02m+c12m*i)*ZMAT^2+(c03m+c13m*i)*ZMAT^3+(c04m+c14m*i)*ZMAT^4+(c05m+c15m*i)*ZMAT^5)*SIGI+
    ((c000+c100*i)*ID3+(c010+c110*i)*ZMAT+(c020+c120*i)*ZMAT^2+(c030+c130*i)*ZMAT^3+(c040+c140*i)*ZMAT^4+(c050+c150*i)*ZMAT^5)+
    ((c00p+c10p*i)*ID3+(c01p+c11p*i)*ZMAT+(c02p+c12p*i)*ZMAT^2+(c03p+c13p*i)*ZMAT^3+(c04p+c14p*i)*ZMAT^4+(c05p+c15p*i)*ZMAT^5)*SIG$
SSSTAR0:=
    SIG*((c00m-c10m*i)*ID3+(c01m-c11m*i)*ZMATI+(c02m-c12m*i)*ZMATI^2+(c03m-c13m*i)*ZMATI^3+(c04m-c14m*i)*ZMATI^4+(c05m-c15m*i)*ZMATI^5)+
    ID3*((c000-c100*i)*ID3+(c010-c110*i)*ZMATI+(c020-c120*i)*ZMATI^2+(c030-c130*i)*ZMATI^3+(c040-c140*i)*ZMATI^4+(c050-c150*i)*ZMATI^5)+
    SIGI*((c00p-c10p*i)*ID3+(c01p-c11p*i)*ZMATI+(c02p-c12p*i)*ZMATI^2+(c03p-c13p*i)*ZMATI^3+(c04p-c14p*i)*ZMATI^4+(c05p-c15p*i)*ZMATI^5)$

SSSTAR:=FFI*SSSTAR0*FF$

% The following is zero:
FF*SSSTAR-sub({i=-i,z=z^(-1)},tp(SS))*FF;


CondMtxDM7Type1:=mat(
(23,1,45,23,1,35,20,1,31,22,0,13,2,46,45,3,47,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,1,45,23,1,35,20,1,31,22,0,13,2,46,45,3,47,16),
(15,0,20,15,0,26,0,0,48,24,2,14,8,1,20,48,48,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,20,15,0,26,0,0,48,24,2,14,8,1,20,48,48,22),
(21,0,42,46,1,5,1,2,17,29,47,42,37,46,20,32,48,42,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,42,46,1,5,1,2,17,29,47,42,37,46,20,32,48,42),
(21,0,7,44,0,45,25,0,20,14,3,25,1,0,26,7,47,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,7,44,0,45,25,0,20,14,3,25,1,0,26,7,47,9),
(28,0,35,29,0,27,34,48,14,1,46,3,21,1,15,17,4,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,35,29,0,27,34,48,14,1,46,3,21,1,15,17,4,29),
(42,0,28,33,0,37,37,2,21,40,48,8,7,47,26,1,48,47,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,42,0,28,33,0,37,37,2,21,40,48,8,7,47,26,1,48,47),
(0,0,39,46,0,45,2,48,31,29,46,32,1,1,20,2,4,44,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,39,46,0,45,2,48,31,29,46,32,1,1,20,2,4,44),
(0,0,6,34,0,22,24,2,15,33,48,6,40,47,2,38,48,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,34,0,22,24,2,15,33,48,6,40,47,2,38,48,24),
(25,1,12,4,1,12,28,45,25,8,48,27,44,2,22,47,4,35,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,1,12,4,1,12,28,45,25,8,48,27,44,2,22,47,4,35),
(23,0,38,30,0,24,38,3,5,36,46,1,6,47,32,28,1,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23,0,38,30,0,24,38,3,5,36,46,1,6,47,32,28,1,6),
(1,0,41,5,48,36,16,47,38,20,4,12,45,3,14,0,44,44,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,41,5,48,36,16,47,38,20,4,12,45,3,14,0,44,44),
(40,0,9,4,2,33,3,46,36,16,48,18,43,1,21,35,3,31,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,40,0,9,4,2,33,3,46,36,16,48,18,43,1,21,35,3,31),
(12,0,3,16,1,43,37,47,44,42,0,45,13,1,37,38,1,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,3,16,1,43,37,47,44,42,0,45,13,1,37,38,1,5),
(10,0,10,7,0,5,33,1,1,21,48,4,21,48,46,11,1,38,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,10,7,0,5,33,1,1,21,48,4,21,48,46,11,1,38),
(15,0,13,1,1,19,24,46,26,30,1,17,6,2,22,31,48,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,15,0,13,1,1,19,24,46,26,30,1,17,6,2,22,31,48,3),
(26,0,9,6,1,41,48,0,42,8,48,46,10,48,28,43,2,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,26,0,9,6,1,41,48,0,42,8,48,46,10,48,28,43,2,29),
(1,1,41,22,46,2,1,2,42,19,1,9,3,0,31,48,47,47,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,41,22,46,2,1,2,42,19,1,9,3,0,31,48,47,47),
(11,0,40,4,1,48,11,47,45,38,0,34,35,2,46,22,48,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,40,4,1,48,11,47,45,38,0,34,35,2,46,22,48,21))$

CondMtxDM3A:=(1/3)*mat(
(24,3,0,18,3,12,24,3,9,21,0,24,18,18,21,15,21,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,3,0,18,3,12,24,3,9,21,0,24,18,18,21,15,21,21),
(12,0,6,15,0,21,15,0,3,15,6,21,21,3,3,15,24,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,6,15,0,21,15,0,3,15,6,21,21,3,3,15,24,15),
(18,0,0,21,3,6,6,6,15,15,21,3,18,18,9,9,24,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,21,3,6,6,6,15,15,21,3,18,18,9,9,24,9),
(9,0,12,15,0,24,24,0,21,24,9,3,6,0,9,0,21,9,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,12,15,0,24,24,0,21,24,9,3,6,0,9,0,21,9),
(19,0,8,17,0,4,15,25,22,24,2,18,1,2,12,21,0,8,25,0,12,26,0,4,24,26,4,1,24,17,18,1,0,12,4,11),
(2,0,15,1,0,23,3,1,23,26,3,10,9,26,0,15,23,16,19,0,8,17,0,4,15,25,22,24,2,18,1,2,12,21,0,8),
(1,0,4,2,0,1,15,0,10,8,25,4,2,0,24,21,2,17,21,0,17,9,0,9,18,2,0,26,26,7,25,25,3,9,26,24),
(6,0,10,18,0,18,9,25,0,1,1,20,2,2,24,18,1,3,1,0,4,2,0,1,15,0,10,8,25,4,2,0,24,21,2,17),
(21,0,12,6,0,24,24,24,15,24,18,24,24,3,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,21,0,12,6,0,24,24,24,15,24,18,24,24,3,0,0,12,0),
(3,0,15,0,0,9,0,6,18,6,24,9,21,21,12,0,24,24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,15,0,0,9,0,6,18,6,24,9,21,21,12,0,24,24),
(3,3,6,12,3,9,9,15,15,3,24,6,18,6,0,12,12,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,6,12,3,9,9,15,15,3,24,6,18,6,0,12,12,3),
(6,0,12,9,0,24,0,9,9,9,18,6,21,21,0,3,3,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,12,9,0,24,0,9,9,9,18,6,21,21,0,3,3,21),
(25,0,23,25,25,18,9,4,23,4,0,21,20,25,23,17,25,1,1,0,19,25,26,0,4,25,13,17,4,10,21,3,11,17,22,22),
(26,0,8,2,1,0,23,2,14,10,23,17,6,24,16,10,5,5,25,0,23,25,25,18,9,4,23,4,0,21,20,25,23,17,25,1),
(1,0,24,13,0,9,20,25,21,21,2,20,19,2,20,11,25,4,15,0,19,9,2,18,8,24,7,26,26,23,11,1,21,18,3,24),
(12,0,8,18,25,9,19,3,20,1,1,4,16,26,6,9,24,3,1,0,24,13,0,9,20,25,21,21,2,20,19,2,20,11,25,4),
(24,0,24,9,21,9,0,15,18,0,24,6,6,18,0,18,24,12,0,0,12,9,6,0,3,15,0,21,0,12,0,6,6,3,6,6),
(0,0,15,18,21,0,24,12,0,6,0,15,0,21,21,24,21,21,24,0,24,9,21,9,0,15,18,0,24,6,6,18,0,18,24,12),
(21,0,15,0,12,18,12,9,9,3,21,6,12,6,24,21,18,21,21,0,18,9,0,18,6,6,9,15,21,9,12,21,12,15,6,9),
(6,0,9,18,0,9,21,21,18,12,6,18,15,6,15,12,21,18,21,0,15,0,12,18,12,9,9,3,21,6,12,6,24,21,18,21),
(18,0,6,21,24,12,15,18,9,3,18,15,15,12,6,24,12,24,6,0,3,12,6,0,24,9,21,18,6,3,24,12,21,3,21,9),
(21,0,24,15,21,0,3,18,6,9,21,24,3,15,6,24,6,18,18,0,6,21,24,12,15,18,9,3,18,15,15,12,6,24,12,24),
(6,0,24,9,18,24,18,18,21,24,6,15,18,18,15,6,0,3,21,0,18,12,6,24,24,0,6,15,21,9,24,21,0,0,12,12),
(6,0,9,15,21,3,3,0,21,12,6,18,3,6,0,0,15,15,6,0,24,9,18,24,18,18,21,24,6,15,18,18,15,6,0,3),
(0,3,21,24,18,6,21,6,3,3,3,24,18,0,6,0,21,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,21,24,18,6,21,6,3,3,3,24,18,0,6,0,21,3),
(9,0,9,3,3,9,12,21,15,3,0,0,12,6,24,9,24,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,9,3,3,9,12,21,15,3,0,0,12,6,24,9,24,18))$

CondMtxDM3B:=(1/3)*mat(
(24,3,0,18,3,12,24,3,9,21,0,24,18,18,21,15,21,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,3,0,18,3,12,24,3,9,21,0,24,18,18,21,15,21,21),
(12,0,6,15,0,21,15,0,3,15,6,21,21,3,3,15,24,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,6,15,0,21,15,0,3,15,6,21,21,3,3,15,24,15),
(18,0,0,12,21,15,15,15,24,24,12,21,18,18,9,9,6,9,0,0,6,0,24,6,6,21,9,24,24,12,12,9,9,18,0,9),
(0,0,21,0,3,21,21,6,18,3,3,15,15,18,18,9,0,18,18,0,0,12,21,15,15,15,24,24,12,21,18,18,9,9,6,9),
(9,0,3,24,0,6,6,0,12,6,9,21,15,0,9,0,12,9,18,0,12,3,6,9,9,12,24,0,24,9,15,9,0,18,15,0),
(9,0,15,24,21,18,18,15,3,0,3,18,12,18,0,9,12,0,9,0,3,24,0,6,6,0,12,6,9,21,15,0,9,0,12,9),
(3,0,24,24,0,12,18,21,12,18,6,0,3,6,9,9,0,24,21,0,9,24,0,12,18,24,12,3,18,24,0,3,0,9,12,6),
(6,0,18,3,0,15,9,3,15,24,9,3,0,24,0,18,15,21,3,0,24,24,0,12,18,21,12,18,6,0,3,6,9,9,0,24),
(3,0,12,6,0,3,18,0,3,24,21,12,6,0,18,9,6,24,9,0,24,0,0,0,0,6,0,24,24,21,21,21,9,0,24,18),
(18,0,3,0,0,0,0,21,0,3,3,6,6,6,18,0,3,9,3,0,12,6,0,3,18,0,3,24,21,12,6,0,18,9,6,24),
(5,0,17,22,0,7,25,25,1,25,21,7,7,2,0,18,8,0,16,0,3,25,0,25,10,20,16,2,7,7,9,7,2,18,0,13),
(11,0,24,2,0,2,17,7,11,25,20,20,18,20,25,9,0,14,5,0,17,22,0,7,25,25,1,25,21,7,7,2,0,18,8,0),
(11,0,1,9,0,6,9,4,12,13,25,15,14,23,26,0,25,7,24,0,4,13,0,10,16,0,7,18,20,1,14,0,13,18,7,17),
(3,0,23,14,0,17,11,0,20,9,7,26,13,0,14,9,20,10,11,0,1,9,0,6,9,4,12,13,25,15,14,23,26,0,25,7),
(3,3,6,12,3,9,9,15,15,3,24,6,18,6,0,12,12,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,6,12,3,9,9,15,15,3,24,6,18,6,0,12,12,3),
(6,0,12,9,0,24,0,9,9,9,18,6,21,21,0,3,3,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,12,9,0,24,0,9,9,9,18,6,21,21,0,3,3,21),
(3,0,6,3,3,0,0,21,6,21,0,9,24,3,6,15,3,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,6,3,3,0,0,21,6,21,0,9,24,3,6,15,3,12),
(12,0,18,21,0,0,24,3,9,9,24,24,12,24,24,24,3,21,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,18,21,0,0,24,3,9,9,24,24,12,24,24,24,3,21),
(17,0,17,3,25,21,0,23,6,9,8,20,11,6,0,15,17,22,9,0,22,12,20,18,19,14,18,7,0,22,9,20,11,10,20,2),
(18,0,5,15,7,9,8,13,9,20,0,5,18,7,16,17,7,25,17,0,17,3,25,21,0,23,6,9,8,20,11,6,0,15,17,22),
(25,0,5,0,4,6,4,21,3,1,25,11,13,2,8,25,6,7,16,0,6,3,0,24,2,20,21,23,7,12,4,7,4,14,20,12),
(11,0,21,24,0,3,25,7,6,4,20,15,23,20,23,13,7,15,25,0,5,0,4,6,4,21,3,1,25,11,13,2,8,25,6,7),
(9,0,3,24,12,6,21,9,18,15,9,21,21,6,3,12,6,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,3,24,12,6,21,9,18,15,9,21,21,6,3,12,6,12),
(3,0,12,18,9,12,9,9,24,12,3,21,9,9,21,3,0,15,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,12,18,9,12,9,9,24,12,3,21,9,9,21,3,0,15),
(0,3,21,24,18,6,21,6,3,3,3,24,18,0,6,0,21,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,21,24,18,6,21,6,3,3,3,24,18,0,6,0,21,3),
(9,0,9,3,3,9,12,21,15,3,0,0,12,6,24,9,24,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,9,0,9,3,3,9,12,21,15,3,0,0,12,6,24,9,24,18))$


;end;

 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generators for $\bar\Gamma$ in the case $(\cC_{20},p=2,\emptyset)$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


b1:= -6/14$ b2:= -4/14$ b3:=  2/14$ b4:=  1/14$ b5:= -4/14$ b6:=  2/14$ b7:=  0/14$ b8:=  0/14$ b9:=  0/14$
b10:= 5/14$ b11:=-6/14$ b12:=-4/14$ b13:=-5/14$ b14:=-8/14$ b15:= 4/14$ b16:= 5/14$ b17:=-6/14$ b18:=-4/14$ 
b19:= 2/14$ b20:= 2/14$ b21:=-4/14$ b22:= 5/14$ b23:= 0/14$ b24:=-6/14$ b25:= 2/14$ b26:= 8/14$ b27:=-6/14$
b28:= 7/14$ b29:=-2/14$ b30:=-4/14$ b31:=-1/14$ b32:=-2/14$ b33:= 0/14$ b34:=-1/14$ b35:= 8/14$ b36:=-8/14$

z1:= 0$ z2:= 0$ z3:= 0$ z4:= 0$ z5:= 1$ z6:= 0$ z7:= 0$ z8:= 0$ z9:= 0$ 
z10:=0$ z11:=0$ z12:=0$ z13:=0$ z14:=0$ z15:=0$ z16:=0$ z17:=0$ z18:=0$
z19:=0$ z20:=0$ z21:=0$ z22:=0$ z23:=0$ z24:=0$ z25:=0$ z26:=0$ z27:=0$
z28:=0$ z29:=0$ z30:=0$ z31:=0$ z32:=0$ z33:=0$ z34:=0$ z35:=0$ z36:=0$

BB:=((b1+b19*i)*ID3+(b4+b22*i)*ZMAT+(b7+b25*i)*ZMAT^2+(b10+b28*i)*ZMAT^3+(b13+b31*i)*ZMAT^4+(b16+b34*i)*ZMAT^5)*SIGI+
    ((b2+b20*i)*ID3+(b5+b23*i)*ZMAT+(b8+b26*i)*ZMAT^2+(b11+b29*i)*ZMAT^3+(b14+b32*i)*ZMAT^4+(b17+b35*i)*ZMAT^5)+
    ((b3+b21*i)*ID3+(b6+b24*i)*ZMAT+(b9+b27*i)*ZMAT^2+(b12+b30*i)*ZMAT^3+(b15+b33*i)*ZMAT^4+(b18+b36*i)*ZMAT^5)*SIG$

% Explicitly,
BB:=(1/28)*mat(
(2*(-3*h*i*z-4*h*i+5*h*z^2+h*z+h+7*i*z^2-7*i*z-7*i+14*z^2+7*z),4*h*(-i*z+i-z^2+z),2*(h*i*z-h*i-3*h*z^2+3*h+7*i*z^2+14*i*z+7*i+7*z-7)),
(5*h*i*z^2-5*h*i+h*z^2+8*h*z+5*h+21*i*z^2-7*i+7*z^2-7,2*(-3*h*i*z^2+3*h*i-4*h*z^2-5*h*z-5*h-14*i*z^2-7*i*z+7*i-7*z^2-14*z-7),2*(-2*h*i*z^2+2*h*i+4*h*z^2+2*h*z+h-7*i)),
(3*h*i*z^2+3*h*i*z+8*h*i-3*h*z^2-6*h*z-5*h-7*i*z^2-14*i*z+7*i-7*z^2-7*z,-5*h*i*z^2-5*h*i*z-4*h*i+7*h*z^2-h*z-6*h-21*i*z^2-21*i*z-14*i-7*z^2-7*z-28,2*(3*h*i*z^2+3*h*i*z+h*i-h*z^2+4*h*z+4*h+7*i*z^2+14*i*z-7*z^2+7*z+7)))$

ZZ:=((z1+z19*i)*ID3+(z4+z22*i)*ZMAT+(z7+z25*i)*ZMAT^2+(z10+z28*i)*ZMAT^3+(z13+z31*i)*ZMAT^4+(z16+z34*i)*ZMAT^5)*SIGI+
    ((z2+z20*i)*ID3+(z5+z23*i)*ZMAT+(z8+z26*i)*ZMAT^2+(z11+z29*i)*ZMAT^3+(z14+z32*i)*ZMAT^4+(z17+z35*i)*ZMAT^5)+
    ((z3+z21*i)*ID3+(z6+z24*i)*ZMAT+(z9+z27*i)*ZMAT^2+(z12+z30*i)*ZMAT^3+(z15+z33*i)*ZMAT^4+(z18+z36*i)*ZMAT^5)*SIG$

% Explicitly, ZZ equals ZMAT. That is,
ZZ:=mat(
(z,  0,  0),
(0,z^2,  0),
(0,  0,z^4))$

% Here is the inverse of BB:
BI:=(1/28)*mat(
(2*(-4*h*i*z^2-2*h*i*z-h*i-5*h*z^2-h*z-h-7*i*z^2+7*i*z+7*i-7),-2*h*i*z^2-5*h*i+10*h*z^2+12*h*z-h+14*i*z^2-28*i*z-35*i+14*z^2+7,2*h*i*z+5*h*i-3*h*z^2-3*h*z-h+7*i*z^2+7*i*z+21*i-14*z-7),
(2*(-h*i*z^2+h*i+2*h*z-2*h-7*z^2-7),2*(2*h*i*z^2+4*h*i*z+h*i+4*h*z^2+5*h*z+5*h+14*i*z^2+7*i*z-7*i+7),2*h*i*z^2+2*h*i*z+3*h*i+2*h*z^2-10*h*z-13*h-42*i*z^2-14*i*z-7*i-14*z^2-14*z+7),
(2*(h*i*z-h*i-8*h*z^2-6*h*z+14*i*z+14*i+7*z+7),2*(h*i*z^2+h*i*z-2*h*i+2*h*z^2-2*h+7*z^2+7*z),2*(2*h*i*z^2-2*h*i*z+h*z^2-4*h*z-4*h-7*i*z^2-14*i*z)))$

% Here is the inverse of ZZ:
ZI:=mat(
(z^6,  0,  0),
(  0,z^5,  0),
(  0,  0,z^3))$

BBSTAR:=sub({i=-i,z=z^6},tp(BB))$
ZZSTAR:=sub({i=-i,z=z^6},tp(ZZ))$

% Checking that the elements $\xi$ satisfy $\iota(\xi)\xi=1$:
% The following are zero:
BBSTAR*FF*BB-FF;
ZZSTAR*FF*ZZ-FF;

bvec:=mat((b1),(b2),(b3),(b4),(b5),(b6),(b7),(b8),(b9),(b10),(b11),(b12),(b13),(b14),(b15),(b16),(b17),(b18),
          (b19),(b20),(b21),(b22),(b23),(b24),(b25),(b26),(b27),(b28),(b29),(b30),(b31),(b32),(b33),(b34),(b35),(b36))$

zvec:=mat((z1),(z2),(z3),(z4),(z5),(z6),(z7),(z8),(z9),(z10),(z11),(z12),(z13),(z14),(z15),(z16),(z17),(z18),
          (z19),(z20),(z21),(z22),(z23),(z24),(z25),(z26),(z27),(z28),(z29),(z30),(z31),(z32),(z33),(z34),(z35),(z36))$

% The following is zero:

CondMtxDM7Type1*bvec - (1/2)*mat(
(-62),(-16),(-5),(-23),(-152),(-26),(-30),(26),(-58),(-10),(-141),(-9),(-24),(-48),(-117),(17),(-77),(-39),(-36),(-6),(-92),(-20),(-78),(-48),(43),(7),(-90),(-48),(-8),(84),(-103),(-135),(-22),(-36),(-15),(1));

CondMtxDM7Type1*zvec - mat((1),(0),(0),(0),(1),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(1),(0),(0),(0),(48),(0),(2),(0),(1),(0),(0),(0),(1),(0),(1),(0),(46),(0),(1),(0));

% Since there are no factors of 3 in the denominators of the b_i's and z_i's, both BB and ZZ satisfy the
% Type 1 3-adic condition.

% Relations. The following are zero, checking that the relations in the stated 
% presentation of group $\bar\Gamma_{(\cC_{20},p=2,\emptyset)}$ do indeed hold:

BB^3 - dt*ID3;
ZZ^7 - ID3;
(BB*ZZ^3*BB*ZI)^3 - dt^2*ID3;
(BB*ZZ^3*BB*ZI^2)^3 - dt^2*ID3;
BB*ZI^3*BB*ZI^3*BB*ZI*BI*ZI^3*BI*ZZ*BB*ZZ^3*BB*ZZ^3 - dt*ID3;
BB*ZZ^3*BB*ZZ*BI*ZI^3*BI*ZZ^3*BI*ZZ^3*BI*ZZ^3*BI*ZZ^3 - dti*ID3;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generators for $\bar\Gamma$ in the case $(\cC_{20},p=2,\{3+\})$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

a1:=-8/21$ a2:=1/3$ a3:=4/21$ a4:=-4/21$ a5:=-2/7$ a6:=-1/7$ a7:=-2/21$ a8:=1/7$ a9:=1/3$
a10:=5/21$ a11:=-1/21$ a12:=-8/21$ a13:=-4/21$ a14:=1/7$ a15:=8/21$ a16:=-1/21$ a17:=1/21$ a18:=-1/21$ 
a19:=4/21$ a20:=-11/21$ a21:=-1/7$ a22:=8/21$ a23:=-5/7$ a24:=1/21$ a25:=5/21$ a26:=-5/21$ a27:=-3/7$ 
a28:=2/21$ a29:=-3/7$ a30:=-5/21$ a31:=2/7$ a32:=-13/21$ a33:=-1/21$ a34:=-4/21$ a35:=-1/7$ a36:=-4/21$ 

b1:=-1/6$ b2:=-2/21$ b3:=-4/21$ b4:=-11/42$ b5:=-1/7$ b6:=2/21$ b7:=-2/7$ b8:=-10/21$ b9:=-10/21$ 
b10:=-17/42$ b11:=-2/21$ b12:=-5/21$ b13:=1/21$ b14:=1/3$ b15:=-4/21$ b16:=-16/21$ b17:=-4/21$ b18:=-1/3$ 
b19:=1/14$ b20:=0$ b21:=1/21$ b22:=1/2$ b23:=-3/7$ b24:=-4/21$ b25:=-1/21$ b26:=8/21$ b27:=-8/21$ 
b28:=-1/14$ b29:=2/21$ b30:=1/7$ b31:=2/21$ b32:=1/21$ b33:=-2/7$ b34:=2/7$ b35:=-2/21$ b36:=0$ 

c1:=-5/21$ c2:=1/21$ c3:=-2/7$ c4:=-4/21$ c5:=2/7$ c6:=-2/3$ c7:=-4/21$ c8:=1/21$ c9:=-1/7$ 
c10:=2/21$ c11:=0$ c12:=-8/21$ c13:=0$ c14:=17/21$ c15:=-1/21$ c16:=-10/21$ c17:=1/7$ c18:=-10/21$ 
c19:=-1/21$ c20:=11/21$ c21:=-16/21$ c22:=1/3$ c23:=0$ c24:=0$ c25:=-4/21$ c26:=3/7$ c27:=-1/21$ 
c28:=1/21$ c29:=10/21$ c30:=2/21$ c31:=1/21$ c32:=1/7$ c33:=-5/21$ c34:=-11/21$ c35:=2/21$ c36:=-8/21$ 


AA:=((a1+a19*i)*ID3+(a4+a22*i)*ZMAT+(a7+a25*i)*ZMAT^2+(a10+a28*i)*ZMAT^3+(a13+a31*i)*ZMAT^4+(a16+a34*i)*ZMAT^5)*SIGI+
    ((a2+a20*i)*ID3+(a5+a23*i)*ZMAT+(a8+a26*i)*ZMAT^2+(a11+a29*i)*ZMAT^3+(a14+a32*i)*ZMAT^4+(a17+a35*i)*ZMAT^5)+
    ((a3+a21*i)*ID3+(a6+a24*i)*ZMAT+(a9+a27*i)*ZMAT^2+(a12+a30*i)*ZMAT^3+(a15+a33*i)*ZMAT^4+(a18+a36*i)*ZMAT^5)*SIG$

BB:=((b1+b19*i)*ID3+(b4+b22*i)*ZMAT+(b7+b25*i)*ZMAT^2+(b10+b28*i)*ZMAT^3+(b13+b31*i)*ZMAT^4+(b16+b34*i)*ZMAT^5)*SIGI+
    ((b2+b20*i)*ID3+(b5+b23*i)*ZMAT+(b8+b26*i)*ZMAT^2+(b11+b29*i)*ZMAT^3+(b14+b32*i)*ZMAT^4+(b17+b35*i)*ZMAT^5)+
    ((b3+b21*i)*ID3+(b6+b24*i)*ZMAT+(b9+b27*i)*ZMAT^2+(b12+b30*i)*ZMAT^3+(b15+b33*i)*ZMAT^4+(b18+b36*i)*ZMAT^5)*SIG$

CC:=((c1+c19*i)*ID3+(c4+c22*i)*ZMAT+(c7+c25*i)*ZMAT^2+(c10+c28*i)*ZMAT^3+(c13+c31*i)*ZMAT^4+(c16+c34*i)*ZMAT^5)*SIGI+
    ((c2+c20*i)*ID3+(c5+c23*i)*ZMAT+(c8+c26*i)*ZMAT^2+(c11+c29*i)*ZMAT^3+(c14+c32*i)*ZMAT^4+(c17+c35*i)*ZMAT^5)+
    ((c3+c21*i)*ID3+(c6+c24*i)*ZMAT+(c9+c27*i)*ZMAT^2+(c12+c30*i)*ZMAT^3+(c15+c33*i)*ZMAT^4+(c18+c36*i)*ZMAT^5)*SIG$

AA:=(1/84)*mat(
(2*(-2*h*i*z^2-h*i*z+3*h*i+6*h*z^2+9*h*z+13*h+28*i*z^2-7*i*z-21*i-21*z+7),2*(-7*h*i*z^2-8*h*i*z+8*h*i+h*z^2+5*h*z+h-7*i*z^2+7*i*z-7*i+7*z^2-28*z-14),9*h*i*z^2+4*h*i*z-6*h*i-9*h*z^2+4*h*z-2*h+21*i*z^2+28*i*z+42*i+21*z^2+28*z-14),
(2*(-h*i*z^2-6*h*i*z-7*h*i+4*h*z^2+6*h*z-3*h+14*i*z^2-7*i+7*z^2-21),2*(h*i*z^2+2*h*i*z+4*h*i+3*h*z^2-6*h*z-4*h-35*i*z^2-28*i*z-14*i-21*z^2+14),2*(-h*i*z^2+7*h*i*z+15*h*i+4*h*z^2-h*z+4*h+14*i*z^2+7*i*z-35*z^2-7*z+7)),
(26*h*i*z^2+19*h*i*z+11*h*i-4*h*z^2+h*z-11*h-28*i*z^2-35*i*z-7*i+14*z^2+49*z+49,2*(-5*h*i*z^2+h*i*z-3*h*i+2*h*z^2-4*h*z-12*h-14*i*z^2-14*i*z-7*z^2-7*z-21),2*(h*i*z^2-h*i*z-7*h*i-9*h*z^2-3*h*z+12*h+7*i*z^2+35*i*z+14*i+21*z^2+21*z+21)))$

BB:=(1/84)*mat(
(2*(2*h*i*z^2-2*h*i*z+7*h*i-4*h*z^2-2*h*z-h+14*i*z^2-14*i*z+7*i-28*z^2-14*z-7),2*(2*h*i*z^2-5*h*i*z-4*h*i-3*h*z^2-3*h*z+6*h-7*i*z^2+7*i*z+14*i+21*z),2*(3*h*i*z^2-9*h*i*z-h*i+3*h*z^2+2*h*z-5*h-21*i*z^2-7*i+21*z^2-7*z+7)),
(-32*h*i*z^2-15*h*i*z+5*h*i-12*h*z^2-15*h*z-h+28*i*z^2+21*i*z+35*i-21*z-49,4*(-2*h*i*z^2-h*i*z-4*h*i+h*z^2+2*h*z-3*h-14*i*z^2-7*i*z-7*i+7*z^2+14*z),2*(-7*h*i*z^2-2*h*i*z-5*h*i+3*h*z+11*h+14*i*z^2+7*i*z+7*i+21*z^2-7)),
(-3*h*i*z^2+17*h*i+8*h*z^2+7*h*z+6*h-21*i*z+14*i-49*z^2-56*z-21,17*h*i*z^2+32*h*i*z+21*h*i-3*h*z^2+12*h*z-9*h-7*i*z^2-28*i*z-21*i-21*z^2+63,2*(2*h*i*z^2+4*h*i*z+h*i+2*h*z^2-2*h*z+7*h+14*i*z^2+28*i*z+7*i+14*z^2-14*z+7)))$

CC:=(1/84)*mat(
(2*(-3*h*i*z^2+17*h*i-8*h*z^2-10*h*z-3*h+35*i-35*z^2-28*z-21),2*(2*h*i*z^2-8*h*i*z-h*i-10*h*z^2-2*h*z+5*h+14*i*z^2+28*i*z-7*i+14*z^2-14*z-7),18*h*i*z^2-4*h*i*z-7*h*i-18*h*z^2+16*h*z+9*h+42*i*z^2+56*i*z+35*i+42*z^2+28*z+21),
(2*(-10*h*i*z^2-12*h*i*z-6*h*i+11*h*z^2+12*h*z+5*h+35*i*z^2-21*i+14*z^2-28),2*(3*h*i*z^2+3*h*i*z+h*i-2*h*z^2+8*h*z+h+7*i+7*z^2+35*z+7),2*(-10*h*i*z^2-2*h*i*z+5*h*i+8*h*z^2+10*h*z+3*h+14*i*z^2-14*i*z-49*i-28*z^2-14*z-21)),
(19*h*i*z^2+29*h*i*z+8*h*i+17*h*z^2-5*h*z+2*h-35*i*z^2-49*i*z-70*i-7*z^2+7*z+56,2*(-2*h*i*z^2+10*h*i*z+6*h*i+h*z^2-11*h*z-18*h-35*i*z^2-35*i*z-14*z^2-14*z),2*(-3*h*i*z+3*h*i+10*h*z^2+2*h*z+2*h+28*z^2-7*z-7)))$

AASTAR:=sub({i=-i,z=z^6},tp(AA))$
BBSTAR:=sub({i=-i,z=z^6},tp(BB))$
CCSTAR:=sub({i=-i,z=z^6},tp(CC))$

% Checking that the elements $\xi$ satisfy $\iota(\xi)\xi=1$:
% The following are zero:
AASTAR*FF*AA-FF;
BBSTAR*FF*BB-FF;
CCSTAR*FF*CC-FF;

% Here are the inverses of AA, BB and CC, respectively:

AI:=FFI*AASTAR*FF$
BI:=FFI*BBSTAR*FF$
CI:=FFI*CCSTAR*FF$

% Explicitly:
AI:=(1/84)*mat(
(2*(9*h*i*z+5*h*i-8*h*z^2+2*h*z+6*h+14*i-14*z^2-7*z+21),2*(2*h*i*z^2-5*h*i*z-4*h*i+2*h*z^2+10*h*z+2*h+14*i*z^2-14*i*z-28*i+14*z^2+7*z+14),-27*h*i*z^2-h*i*z+14*h*i+9*h*z^2-11*h*z-12*h-21*i*z^2-49*i*z-28*i-63*z^2-77*z-42),
(2*(8*h*i*z^2+18*h*i*z+9*h*i-4*h*z^2-6*h*z-4*h-28*i*z^2-28*z^2+35),2*(9*h*i*z^2-2*h*i+10*h*z^2+8*h*z+10*h-14*i+7*z^2+14*z+28),2*(-7*h*i*z^2-2*h*i*z+2*h*i+8*h*z^2-2*h*z-6*h-28*i*z^2-14*i*z-28*i-7*z^2-14*z)),
(4*h*i*z^2+14*h*i*z+17*h*i-22*h*z^2-26*h*z-h-14*i*z^2+14*i*z+35*i-28*z^2-14*z+35,2*(10*h*i*z^2-8*h*i*z-9*h*i-2*h*z^2+4*h*z+12*h+28*i*z^2+28*i*z+28*z^2+28*z+21),2*(-9*h*i*z^2-9*h*i*z-3*h*i-2*h*z^2-10*h*z+5*h+21*i+7*z^2-7*z-7)))$

BI:=(1/84)*mat(
(2*(17*h*i*z^2+10*h*i*z+h*i-7*h*z^2+10*h*z+11*h+35*i*z^2+28*i*z+7*i-7*z^2+28*z+35),-23*h*i*z^2-19*h*i*z+7*h*i-27*h*z^2+9*h*z-3*h+7*i*z^2+35*i*z+7*i+21*z^2-63*z-21,-6*h*i*z+13*h*i+9*h*z^2+h*z-10*h-21*i*z^2-21*i*z-14*i-14*z-7),
(2*(-4*h*i*z^2+4*h*i-3*h*z^2-6*h*z-5*h-7*i*z^2+7*i-14),2*(-7*h*i*z^2-17*h*i*z-11*h*i+17*h*z^2+7*h*z-3*h-7*i*z^2-35*i*z-35*i+35*z^2+7*z-21),4*h*i*z^2+23*h*i*z+29*h*i+36*h*z^2+27*h*z+7*h+28*i*z^2-7*i*z-91*i-84*z^2-21*z+49),
(2*(15*h*i*z^2+9*h*i*z+4*h*i+h*z^2-10*h*z-12*h-21*i*z^2-42*i*z-14*i+7*z^2+35*z+42),2*(4*h*i*z^2+4*h*i*z+6*h*i-3*h*z^2+3*h*z+7*i*z^2+7*i*z),2*(-10*h*i*z^2+7*h*i*z+10*h*i-10*h*z^2-17*h*z-8*h-28*i*z^2+7*i*z+28*i-28*z^2-35*z-14)))$

CI:=(1/84)*mat(
(2*(16*h*i*z^2+11*h*i*z-6*h*i-6*h*z^2+3*h*z+10*h+28*i*z^2+35*i*z+21*z+28),2*(5*h*i*z^2+7*h*i*z+2*h*i-5*h*z^2+11*h*z+h+35*i*z^2+7*i*z-7*i+7*z^2-7*z+28),18*h*i*z^2-11*h*i*z-21*h*i+13*h*z+h+7*i*z-21*i+42*z^2+49*z+7),
(2*(-10*h*i*z^2-12*h*i*z-13*h*i+4*h*z^2+3*h+14*i*z^2-7*i+28*z^2-21),2*(-5*h*i*z^2-16*h*i*z-14*h*i+9*h*z^2+6*h*z-h+7*i*z^2-28*i*z-35*i+21*z^2-28),2*(2*h*i*z^2-5*h*i*z+3*h*i+16*h*z^2+5*h*z-14*h-28*i*z^2-35*i*z-42*i-14*z^2-7*z+7)),
(-7*h*i*z^2+4*h*i*z-4*h*i-13*h*z^2-38*h*z-26*h-49*i*z^2-14*i*z+14*i+35*z^2+28*z+28,4*(-h*i*z^2+5*h*i*z+3*h*i-2*h*z^2-2*h*z-3*h-7*i*z^2-7*i*z-14*z^2-14*z),2*(-11*h*i*z^2+5*h*i*z-h*i-3*h*z^2-9*h*z-9*h-35*i*z^2-7*i*z-7*i-21*z^2-21*z-21)))$

% Relations. The following are zero, checking that the relations in the stated 
% presentation of group $\bar\Gamma_{(\cC_{20},p=2,\{3+\})}$ do indeed hold:

BB^3 - dt*ID3;
AI^2*BB*CC^2*BB*CC^2*AA*CI*BB - dt*ID3;
CC*AA*BB*AI*BI*AI*BI*AI*BI*AI*CI*BI*AI - dti*ID3;
AI*CC*AA*CC^2*AA*CI*AA*BB*CC*AI*CI^2*BI - ID3;
CI*AA*BB*CC*AI*CI*AA*CC*AI*CI^2*BI*AA^2 - ID3;
AA*BI*CC^2*AI*CI^2*BI*AA^3*BB*AA*BB - ID3;
BB*CC^2*AA*CI^3*BI*AA^2*CC^2*AA*CI - ID3;
(BB*CC^3*AI)^3 - dt*ID3;
CI*AA*CI*BB*AI^2*CC*AI*CI^2*AI*CI^3*BI*AA - ID3;
AI*CC*AI*CI^2*BB*AI*CC*AI*CI^3*BI*AI*CC*AI - ID3;
AI*CI*AA*BB*CI*BI*AI*CC*AI*BB*CC*BB*AI*BB*CC*AI*CI - dt*ID3;
BB*CI*BI*AI*CC*AI*BB*CC*AI*CI*BI*AI*CI*BI*AI*CC*AI*BB*CC^2 - ID3;
AI*CC*AI^2*CI^2*BI*AA^2*CC*AA*BB*AI^2*CC*AI*CI^2*AI^2 - ID3;
(BB*AI*BI*AI^2*BB*CC)^3 - dt*ID3;
CI*BI*AI*CI^2*BI*AA^2*CC*AA*CI*BI*AA*BI*AI*CI*BI*AI*BI*AI*CC*AA - dti^2*ID3;
BB*AI*CI*AA*CI*BB*AI*BB*AA*BB*CC*AI*CI*AA*BB*CI*BI*AI*BI*AI*CC*AA*CC - dt*ID3;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generators for $\bar\Gamma$ in the case $(\cC_{20},p=2,\{3-\})$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


AA:=(1/84)*mat(
((4*h+56)*i*z^2+((8*h+28)*i+12*h)*z+(2*h-14)*i+30*h+42,
((-4*h-14)*i-10*h+28)*z^2+(-28*h+28)*i*z+(4*h+14)*i+10*h-28,
((4*h+56)*i+(8*h+28))*z^2+((3*h+21)*i+(15*h+63))*z+(-14*h+14)*i+26*h-14),
(((8*h-14)*i+(14*h+28))*z^2+((-10*h-56)*i+(8*h-14))*z-12*h*i-8*h-28,
((4*h-28)*i+12*h)*z^2+(-4*h-56)*i*z-42*i+2*h+28,
((-24*h+42)*i+(10*h-28))*z^2+((4*h+14)*i+(10*h-28))*z+(20*h-14)*i+22*h-28),
(((21*h-21)*i+(7*h-49))*z^2+((25*h-49)*i+(3*h-21))*z+(31*h-7)*i-3*h+21,
((-18*h-42)*i-6*h-42)*z^2+((-8*h+14)*i-14*h-28)*z+(-2*h+56)*i-8*h-70,
((-8*h-28)*i-12*h)*z^2+((-4*h+28)*i-12*h)*z+(-2*h+14)*i+10*h+14))$

BB:=(1/28)*mat(
(((-2*h-14)*i-6*h-14)*z^2+(-2*h*i-8*h-14)*z+(4*h+28)*i,
-4*h*z^2+(-4*h*i+4*h)*z+4*h*i,
((-6*h-14)*i+(2*h+14))*z^2+(-14*i-2*h)*z-8*h*i),
(((h-7)*i-3*h-21)*z^2+((h+7)*i-5*h-21)*z+(5*h+21)*i+h+7,
(14*i-2*h)*z^2+((2*h+14)*i+(6*h+14))*z+(-2*h+14)*i+10*h+14,
(-4*h*i+8*h)*z^2+4*h*z+(4*h-14)*i+2*h),
(((3*h-7)*i-3*h-7)*z^2+((3*h-14)*i-6*h-7)*z+(8*h+7)*i-5*h,
(14*i-2*h)*z^2+((-h+7)*i+(3*h+21))*z+(-6*h+14)*i+6*h+14,
(2*h*i+(8*h+14))*z^2+(-14*i+2*h)*z-2*h*i+4*h+14))$

CC:=(1/84)*mat(
(((-8*h-28)*i-16*h-56)*z^2+((8*h-14)*i-2*h-28)*z+(14*h+28)*i+4*h+14,
((-20*h-28)*i-12*h+84)*z^2+((-40*h+28)*i-8*h-28)*z+(-10*h-14)*i+6*h-42,
((6*h+42)*i-2*h+14)*z^2+(42*i+6*h)*z+(15*h+21)*i+31*h+35),
(((-8*h+14)*i+6*h)*z^2+((-8*h-28)*i+12*h)*z+(16*h+14)*i+10*h-28,
((16*h+14)*i+(14*h+28))*z^2+((8*h+28)*i+(16*h+56))*z+(-10*h-14)*i+26*h+70,
((-20*h+56)*i+(4*h-112))*z^2+((20*h+28)*i+(12*h-84))*z+(42*h-42)*i+26*h-14),
(((37*h-7)*i+(13*h-49))*z^2+((43*h-49)*i+(11*h+49))*z+(25*h-49)*i+11*h+91,
(-42*i+6*h)*z^2+((8*h-14)*i-6*h)*z+(20*h+28)*i,
((-8*h+14)*i+(2*h+28))*z^2+((-16*h-14)*i-14*h-28)*z+(-4*h+28)*i+12*h))$


AASTAR:=sub({i=-i,z=z^6},tp(AA))$
BBSTAR:=sub({i=-i,z=z^6},tp(BB))$
CCSTAR:=sub({i=-i,z=z^6},tp(CC))$

% Checking that the elements $\xi$ satisfy $\iota(\xi)\xi=1$:
% The following are zero:
AASTAR*FF*AA-FF;
BBSTAR*FF*BB-FF;
CCSTAR*FF*CC-FF;

% Here are the inverses of AA, BB and CC, respectively:

AI:=FFI*AASTAR*FF$
BI:=FFI*BBSTAR*FF$
CI:=FFI*CCSTAR*FF$

% Explicitly:
AI:=(1/84)*mat(
(2*(-3*h*i*z^2+3*h*i-8*h*z^2-4*h*z+5*h+21*i-7*z^2-14*z+7),2*(h*i*z^2-6*h*i*z+5*h*i-4*h*z^2+11*h*z+7*h+14*i*z^2+21*i*z-35*i+7*z^2-14*z-7),-7*h*i*z^2-3*h*i*z+17*h*i-h*z^2-15*h*z-19*h-35*i*z^2-21*i*z+7*i-35*z^2-63*z-35),
(2*(-2*h*i*z^2+7*h*i*z+2*h*i-7*h*z^2-4*h*z-3*h-7*i*z^2+14*i*z+7*i-14*z^2+7*z),2*(3*h*i*z^2+3*h*i*z+h*i+4*h*z^2+8*h*z+9*h-7*i-7*z^2+7*z+21),2*(-7*h*i*z^2-h*i*z+8*h*i+15*h*z^2+4*h*z+2*h+7*i*z^2-14*i*z-56*i-21*z^2-7*z-14)),
(16*h*i*z^2+21*h*i*z+12*h*i-6*h*z^2-19*h*z-10*h-70*i*z^2-63*i*z-42*i+7*z+28,2*(9*h*i*z^2+2*h*i*z-4*h*i+3*h*z^2+7*h*z+4*h+21*i*z^2+7*i*z-14*i+21*z^2+14*z+14),2*(-3*h*i*z-4*h*i+4*h*z^2-4*h*z+7*h+7*i+14*z^2+7*z+14)))$

BI:=(1/28)*mat((2*(4*h*i*z^2+3*h*i*z+h*z^2+h*z+5*h+7*i*z^2+7*i*z-7*i+7*z+14),-5*h*i*z^2-8*h*i*z-h*i-h*z^2-8*h*z-5*h-35*i*z^2-7*i+7*z^2-21,2*h*i*z+5*h*i-3*h*z^2-3*h*z-h+7*i*z^2+7*i*z+21*i-14*z-7),
(2*(-h*i*z^2+h*i+2*h*z-2*h-7*z^2-7),2*(-h*i*z^2-4*h*i*z-2*h*i-h*z+h-7*i*z-7*i+7*z^2),-3*h*i*z^2+5*h*i*z+5*h*i-7*h*z^2+h*z+13*h+35*i*z^2+35*i*z+7*i-7*z^2-7*z-7),
(2*(3*h*i*z^2+2*h*i*z+2*h*i+3*h*z^2+7*h*z+4*h+7*i*z^2-7*i*z-14*i-7*z^2),2*(h*i*z^2+h*i*z-2*h*i+2*h*z^2-2*h+7*z^2+7*z),2*(-3*h*i*z^2+h*i*z+2*h*i-h*z^2+h-7*i*z^2-7*i-7*z^2-7*z)))$

CI:=(1/84)*mat(
(2*(8*h*i*z^2+h*i*z-2*h*i+10*h*z^2+11*h*z+14*h+28*i*z^2-7*i*z-28*i+14*z^2+7*z+28),2*(5*h*i*z^2+13*h*i*z+10*h*i-9*h*z^2-h*z+3*h+7*i*z^2+35*i*z+35*i-21*z^2+7*z),
-9*h*i*z^2-33*h*i*z-21*h*i-11*h*z^2-15*h*z-23*h-105*i*z^2-21*i*z-21*i+35*z^2-21*z-49),
(2*(-16*h*i*z^2-h*i*z-4*h*i-12*h*z^2-9*h*z+14*h+28*i*z^2+49*i*z+28*i-21*z-14),2*(-7*h*i*z^2-8*h*i*z+h*i+h*z^2-10*h*z-5*h-35*i*z^2-28*i*z-7*i-7*z^2-14*z-7),2*(8*h*i*z^2-5*h*i*z-3*h*i+8*h*z^2+9*h*z+4*h+28*i*z^2-7*i*z+28*z^2+21*z-7)),
(-23*h*i*z^2-26*h*i*z-14*h*i+19*h*z^2+2*h*z-28*h-49*i*z^2-70*i*z-28*i+35*z^2-14*z-98,2*(15*h*i*z^2+16*h*i*z+4*h*i+3*h*z^2+12*h*z+6*h+21*i*z^2-28*i*z-28*i-21*z^2+42),
2*(-h*i*z^2+7*h*i*z+h*i-11*h*z^2-h*z+12*h+7*i*z^2+35*i*z+14*i-7*z^2+7*z+21)))$

% Relations. The following are zero, checking that the relations in the stated 
% presentation of group $\bar\Gamma_{(\cC_{20},p=2,\{3-\})}$ do indeed hold:

BB*AA*BI*CC^2*AI - ID3;
(CC*BI*AA)^3 - dti*ID3;
(BI*CC*AA*CI^2)^3 - dti*ID3;
BI*AA*BB*AA*BI*AA*CC*AA*BI^2*CC*AA*CI^2*AA*CC*AA - dti*ID3;
AI*CC^2*AI*CI*BB*AA*CC*AA*CI*BI*CC*BB*AA*BI*AA*CC - ID3;
CI*AI*CC^2*AI^2*BB*AI*CI*AI*BB*AI*BI*CI*BB^2*AI - dt*ID3;
AA*CI*BB*AA*BI*AA*CC*BI*CI*BI*CC*AA*CI^2*AA*CC*BI*CI - dti*ID3;
BI*CI*BB*CC*BI*AA*CC*AI*BI*CC*AI*CC^2*AA*CI^2*AA*CC*BI*CI - dti*ID3;
CC*BI*AA*CI*BB*AI^2*CI*AI*BB*AI*BI*AI*BI*AI*BB*AI*BI*CI*BB*CC - ID3;
AI*CI*AI*CC^2*BI*AA*BB*AA*CI*AI*BB*CI*BI*AA*CC*AI*BI*CI*BB*AI - ID3;
AA*BB*CC^2*AI*CI*BB*AA*BB^2*AI*BI*CI*BB*AA*BB^2*AI*BI*CI*BB - dt^2*ID3;
BB*CC^2*AI*CI*BB*AA*BB*AA*CI*AI*BB*AI*BI*CI*BB*CI*BI*CC*BB*AA*BI*AA - dt*ID3;
BI*CI*BB*CC^2*BI*AA*CC*AI*BI*CC*AI*BI*CC*AI*BI*AI*BI*CC*AA*CI^2*BI*CC - dti^2*ID3;
BB^2*AI*BI*CI*AI*BB*AI*BI*CI*BB*CC*BI*CC*BB^2*AA*CI*AI*BB*AI*BI*CC*AI - dt*ID3;
(BI*AA*BB*AA*CI*AI*BB*CI)^3 - dt*ID3;
CI*AA*BI*CC*BB*AA*BI*AA*CC*BI^2*CC*BB*AA*BI*AA*BB*AA*CI*BB*AA*CI*AI*BB*CI - ID3;
CC^2*BI*AA*CC*AI*BI*CC*BI^2*AA*CI*BB*AA*BI*AA*CC*AI*BI*AI*BI^2*CC*AI*BB - dti^2*ID3;
CC*AI*CI*AI*BI*CC*BB*CI*AI*BI*CC*AA*CI^2*AA*CC*AA*CI*BI*AA*CI*AA*CI^2*AA*CC*BI - dti*ID3;
CC*AI*CI*AI*BI*CC*BB*CI*AI*CC^2*BI*AA^2*CI^2*AA*CC*AA*CI*AI*BB*AI*BI*CC*AI*BB - ID3;