let Z^3=3*Z^2+2$
phiZ :=(i+3-(4*i+1)*Z+i*Z^2)/2$
phi2Z:=(-i+3+(4*i-1)*Z-i*Z^2)/2$

ZMAT:=mat((Z,   0,    0),
          (0,phiZ,    0),
          (0,   0,phi2Z))$

SIGMAT:=mat((0,1,0),
            (0,0,1),
            (5,0,0))$

FF:=mat((5,0,0),
        (0,0,1),
        (0,1,0))$

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

gendt:=(3+4*i)/5$
gendti:=(3-4*i)/5$

;end;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generators for $\bar\Gamma$ in the case $(a=1,p=5,\emptyset)$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Firstly, the sequences of coefficients:
a1:=-24/18$
a2:=9/18$
a3:=0/18$
a4:=3/18$
a5:=-3/18$
a6:=0/18$
a7:=3/18$
a8:=0/18$
a9:=0/18$
a10:=-4/18$
a11:=1/18$
a12:=-4/18$
a13:=19/18$
a14:=-4/18$
a15:=-2/18$
a16:=-7/18$
a17:=1/18$
a18:=2/18$

b1:=-25/90$
b2:=-5/90$
b3:=17/90$
b4:=10/90$
b5:=-25/90$
b6:=7/90$
b7:=5/90$
b8:=10/90$
b9:=-4/90$
b10:=-25/90$
b11:=55/90$
b12:=11/90$
b13:=55/90$
b14:=-10/90$
b15:=16/90$
b16:=-10/90$
b17:=-5/90$
b18:=-7/90$

c1:=13/18$
c2:=-7/18$
c3:=-5/18$
c4:=-1/18$
c5:=-14/18$
c6:=-4/18$
c7:=-2/18$
c8:=5/18$
c9:=1/18$
c10:=3/18$
c11:=15/18$
c12:=3/18$
c13:=0/18$
c14:=-21/18$
c15:=-3/18$
c16:=-3/18$
c17:=6/18$
c18:=0/18$

% Here are the matrix forms of the elements of $\cD$ corresponding
% to the above sequences of coefficients:

AA:=a1*ZMAT^0*SIGMAT^0+a2*ZMAT^0*SIGMAT^1+a3*ZMAT^0*SIGMAT^2+
a4*ZMAT^1*SIGMAT^0+a5*ZMAT^1*SIGMAT^1+a6*ZMAT^1*SIGMAT^2+
a7*ZMAT^2*SIGMAT^0+a8*ZMAT^2*SIGMAT^1+a9*ZMAT^2*SIGMAT^2+
a10*i*ZMAT^0*SIGMAT^0+a11*i*ZMAT^0*SIGMAT^1+a12*i*ZMAT^0*SIGMAT^2+
a13*i*ZMAT^1*SIGMAT^0+a14*i*ZMAT^1*SIGMAT^1+a15*i*ZMAT^1*SIGMAT^2+
a16*i*ZMAT^2*SIGMAT^0+a17*i*ZMAT^2*SIGMAT^1+a18*i*ZMAT^2*SIGMAT^2$

BB:=b1*ZMAT^0*SIGMAT^0+b2*ZMAT^0*SIGMAT^1+b3*ZMAT^0*SIGMAT^2+
b4*ZMAT^1*SIGMAT^0+b5*ZMAT^1*SIGMAT^1+b6*ZMAT^1*SIGMAT^2+
b7*ZMAT^2*SIGMAT^0+b8*ZMAT^2*SIGMAT^1+b9*ZMAT^2*SIGMAT^2+
b10*i*ZMAT^0*SIGMAT^0+b11*i*ZMAT^0*SIGMAT^1+b12*i*ZMAT^0*SIGMAT^2+
b13*i*ZMAT^1*SIGMAT^0+b14*i*ZMAT^1*SIGMAT^1+b15*i*ZMAT^1*SIGMAT^2+
b16*i*ZMAT^2*SIGMAT^0+b17*i*ZMAT^2*SIGMAT^1+b18*i*ZMAT^2*SIGMAT^2$

CC:=c1*ZMAT^0*SIGMAT^0+c2*ZMAT^0*SIGMAT^1+c3*ZMAT^0*SIGMAT^2+
c4*ZMAT^1*SIGMAT^0+c5*ZMAT^1*SIGMAT^1+c6*ZMAT^1*SIGMAT^2+
c7*ZMAT^2*SIGMAT^0+c8*ZMAT^2*SIGMAT^1+c9*ZMAT^2*SIGMAT^2+
c10*i*ZMAT^0*SIGMAT^0+c11*i*ZMAT^0*SIGMAT^1+c12*i*ZMAT^0*SIGMAT^2+
c13*i*ZMAT^1*SIGMAT^0+c14*i*ZMAT^1*SIGMAT^1+c15*i*ZMAT^1*SIGMAT^2+
c16*i*ZMAT^2*SIGMAT^0+c17*i*ZMAT^2*SIGMAT^1+c18*i*ZMAT^2*SIGMAT^2$

% Explicitly:
AA:=mat(
((-7*i*z^2+19*i*z-4*i+3*z^2+3*z-24)/18,(i*z^2-4*i*z+i-3*z+9)/18,(i*(z^2-z-2))/9),
((5*(-i*z^2+i*z+2*i-3*z^2+9*z))/18,(11*i*z^2-35*i*z-4*i+3*z^2-9*z-12)/18,(-i*z^2+4*i*z-i+3)/9),
((5*(i*z^2-4*i*z+i+3*z+3))/18,(5*(-i*z^2+i*z+2*i+3*z^2-9*z))/18,(-2*i*z^2+8*i*z-5*i-3*z^2+3*z)/9))$

BB:=mat(
((-2*i*z^2+11*i*z-5*i+z^2+2*z-5)/18,(-i*z^2-2*i*z+11*i+2*z^2-5*z-1)/18,(-7*i*z^2+16*i*z+11*i-4*z^2+7*z+17)/90),
((-i*z^2+4*i*z+5*i+8*z^2-17*z+5)/18,(2*i*z^2-8*i*z+2*i-z^2+4*z-1)/9,(i*z^2-i*z+i+z^2-4*z+1)/9),
((5*(-i*z^2+4*i*z+5*i-4*z^2+13*z-1))/18,(4*i*z^2-10*i*z+i-2*z^2+5*z+7)/9,(-2*i*z^2+5*i*z+i+z^2-10*z+7)/18))$

CC:=mat(
((-3*i*z^2+3*i-2*z^2-z+13)/18,(6*i*z^2-21*i*z+15*i+5*z^2-14*z-7)/18,(-3*i*z+3*i+z^2-4*z-5)/18),
((5*(3*i*z-3*i+z^2-4*z-5))/18,(-3*i*z^2+15*i*z-12*i+7*z^2-19*z+4)/18,(3*i*z+3*i-2*z^2+2*z+1)/9),
((5*(-6*i*z^2+15*i*z+15*i-z^2+10*z-13))/18,(5*(-z^2+4*z-4))/9,(6*i*z^2-15*i*z-9*i-5*z^2+20*z+1)/18))$


AASTAR:=sub(i=-i,tp(AA))$
BBSTAR:=sub(i=-i,tp(BB))$
CCSTAR:=sub(i=-i,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;

% Checking the integrality conditions:

condmatq2type1:=mat(
(1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0),
(0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0),
(0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0),
(0,0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2),
(1,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0,0),
(0,1,0,0,3,0,0,0,0,0,0,0,0,1,0,0,2,0),
(0,1,0,0,3,0,0,0,0,0,0,0,0,3,0,0,2,0),
(0,0,1,0,0,3,0,0,0,0,0,0,0,0,3,0,0,2),
(1,0,0,3,0,0,0,0,0,0,0,0,3,0,0,2,0,0),
(0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0),
(0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0),
(0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1),
(0,0,0,0,0,3,0,0,2,0,0,1,0,0,3,0,0,0),
(0,0,0,3,0,0,2,0,0,1,0,0,3,0,0,0,0,0),
(0,0,0,0,3,0,0,2,0,0,1,0,0,3,0,0,0,0),
(0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,0,0),
(0,0,0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,0),
(0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,0,0,0))$

condmatq3type1:=mat(
(1,2,4,8,4,5,1,2,7,0,6,0,0,3,0,0,0,3),
(0,6,6,3,0,6,3,3,0,0,0,0,0,0,0,0,0,0),
(0,0,3,0,6,3,3,3,3,0,3,0,0,0,3,0,6,3),
(0,2,0,0,8,4,2,1,6,0,6,3,0,7,7,0,7,7),
(1,2,4,8,7,2,1,5,4,0,6,3,0,6,6,0,0,3),
(0,8,8,1,6,2,4,1,0,0,4,4,0,2,2,0,7,1),
(0,1,4,2,0,1,8,8,6,0,4,1,0,5,5,0,4,7),
(0,0,6,0,3,6,6,6,6,0,3,0,0,6,0,0,3,0),
(1,5,1,5,7,2,7,2,7,0,6,6,0,0,3,0,0,3),
(0,3,0,0,6,0,0,0,6,1,2,4,8,4,5,1,2,7),
(0,0,0,0,0,0,0,0,0,0,6,6,3,0,6,3,3,0),
(0,6,0,0,0,6,0,3,6,0,0,3,0,6,3,3,3,3),
(0,3,6,0,2,2,0,2,2,0,2,0,0,8,4,2,1,6),
(0,3,6,0,3,3,0,0,6,1,2,4,8,7,2,1,5,4),
(0,5,5,0,7,7,0,2,8,0,8,8,1,6,2,4,1,0),
(0,5,8,0,4,4,0,5,2,0,1,4,2,0,1,8,8,6),
(0,6,0,0,3,0,0,6,0,0,0,6,0,3,6,6,6,6),
(0,3,3,0,0,6,0,0,6,1,5,1,5,7,2,7,2,7))$


avec:=mat((a1),(a2),(a3),(a4),(a5),(a6),(a7),(a8),(a9),(a10),(a11),(a12),(a13),(a14),(a15),(a16),(a17),(a18))$

bvec:=mat((b1),(b2),(b3),(b4),(b5),(b6),(b7),(b8),(b9),(b10),(b11),(b12),(b13),(b14),(b15),(b16),(b17),(b18))$

cvec:=mat((c1),(c2),(c3),(c4),(c5),(c6),(c7),(c8),(c9),(c10),(c11),(c12),(c13),(c14),(c15),(c16),(c17),(c18))$

% The following are zero, checking that condmatq2type1*avec, condmatq2type1*bvec and condmatq2type1*cvec
% have entries in $\Z_2$:
condmatq2type1*avec  -  (1/9)*mat((-9),(3),(0),(1),(-5),(-1),(-5),(-1),(14),(4),(-1),(-2),(-5),(34),(-10),(-7),(-5),(31));
condmatq2type1*bvec  - (1/45)*mat((-5),(-10),(10),(20),(20),(-50),(-60),(36),(75),(10),(20),(10),(36),(90),(-15),(10),(29),(80));
condmatq2type1*cvec  -  (1/9)*mat((5),(-8),(-4),(-10),(2),(-29),(-50),(-13),(2),(0),(0),(0),(-8),(-2),(-40),(-26),(-4),(-1));

% The following are zero, checking that condmatq3type1*avec, condmatq3type1*bvec and condmatq3type1*cvec
% have entries in $\Z_3$:
condmatq3type1*avec  -  (1/2)*mat((1),(8),(0),(-3),(-4),(6),(3),(-1),(2),(14),(1),(0),(-2),(14),(-4),(2),(-3),(6));
condmatq3type1*bvec  - (1/10)*mat((36),(21),(13),(15),(41),(26),(42),(25),(34),(39),(68),(-3),(10),(53),(48),(16),(-5),(47));
condmatq3type1*cvec  -  (1/2)*mat((-7),(-10),(-3),(-18),(-17),(-14),(-4),(-15),(-4),(-16),(11),(-18),(-25),(-20),(-18),(-8),(-11),(-15));

% The following are zero, checking the statements about determinants:
det(AA)-1;
det(BB)-(3+4*i)/5;
det(CC)-1;

% Checking that the relations in the group presentation are indeed satisfied:

BBI:=gendti*BB*BB$
AAI:=AA^(-1)$
CCI:=CC^(-1)$

% The following are zero:
BB^3 - gendt*ID3;
CCI*BBI*AA*BBI*AA*BBI*CCI - gendti*ID3;
BB*AA*CC*BB*AAI*BBI*CCI*BBI*CCI*BBI*CCI*AAI*BBI*CCI*BBI*AAI*CC - gendti*ID3;
CCI*AAI*BB*CCI*BBI*AA*CCI*AA*BB*CC*BB*AA*CC*BB*AAI*BBI*CC*BB*AAI - gendt*ID3;
AAI*CC*BBI*AA*BB*AA*BB*AA*CC*BB*AAI*CC*BB*AAI*BBI*AA*CC*BB*AAI - gendt*ID3;
BBI*AAI*CC*AAI*BB*CC*BBI*AA*CC*BB*AAI*CCI*BBI*CCI*AAI*BBI*CCI*BBI*AAI - gendti*ID3;
CC*AAI*BB*CC*BBI*AA*CC*BB*AAI^2*BBI*AAI*BBI*AAI*BBI*AA*CCI*BBI*AA - gendti*ID3;
CCI*AAI*BB*CC*BB*AAI*BBI*AA*CCI*BBI*AA*BB*CCI*BBI*CCI*AAI*BBI*CC*BB*AAI - ID3;
CC*BB*CC*BB*AA*BBI*CCI*AAI*BB*AA*BBI*CC*AA*BBI*CCI*AA*BBI*CCI*BB*AA - ID3;
BB*AA*BB*CC*BB*AA*CC*AA*BBI*AAI*BBI*AAI*BBI*AAI*BB*CCI*AA*CC*BBI*AA - ID3;
CC*BBI*AA*BB*AA*BB*CC*BBI*AA*BB*AA*BB*AA^2*BBI*CCI*BB*AA*CC*BB*AAI - gendt*ID3;
CC*BB*AA*CCI*BBI*AA*CC*AAI*BB*CCI*BBI*AA*BB*AAI*BBI*AA*CC*BB*AAI*BBI*CCI*AAI - ID3;
BB*AA*BB*AA*CC*BBI*CC*AAI*BB*CC*BBI*AA*CCI*BBI*AA^2*CC*BB*AAI*CC*BBI*AA - ID3;
CCI*BBI*CCI*AAI*BB*AA*BB*AA*BB*AA*CCI*AAI*BBI*CCI*AA*BBI*CCI*AAI*BBI*CC*BB*AAI - ID3;
AAI*BBI*AAI*BBI*AAI*BBI*AA*CC*BB*AAI*BBI*CCI*AAI*CC*BBI*AA*BB*AA*BB*AA^2*BBI*CCI - gendti*ID3;
CC*BB*AAI^2*BBI*AAI*BBI*AAI^2*BB*CC*AAI*BBI*CCI*BBI*AAI*BBI*AAI*CC*BBI*AA*BB*AA - gendti*ID3;
AAI*CC*BBI*AA*BB*AA*BB*AA*CCI^2*BB*CCI*AA*BBI*CCI*AAI*BBI*CC*BB*AAI*CCI*AAI*BBI*CCI - ID3;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The generators for $\bar\Gamma$ in the case $(a=1,p=5,\{2\})$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

ad1:=6/36$
ad2:=6/36$
ad3:=12/36$
ad4:=-12/36$
ad5:=45/36$
ad6:=0/36$
ad7:=6/36$
ad8:=-15/36$
ad9:=0/36$
ad10:=14/36$
ad11:=-32/36$
ad12:=-4/36$
ad13:=10/36$
ad14:=11/36$
ad15:=16/36$
ad16:=-4/36$
ad17:=1/36$
ad18:=-4/36$

bd1:=-70/180$
bd2:=-20/180$
bd3:=14/180$
bd4:=-140/180$
bd5:=-85/180$
bd6:=22/180$
bd7:=50/180$
bd8:=25/180$
bd9:=-4/180$
bd10:=-210/180$
bd11:=30/180$
bd12:=-18/180$
bd13:=30/180$
bd14:=-45/180$
bd15:=-24/180$
bd16:=0/180$
bd17:=15/180$
bd18:=18/180$

cd1:=-24/18$
cd2:=9/18$
cd3:=0/18$
cd4:=3/18$
cd5:=-3/18$
cd6:=0/18$
cd7:=3/18$
cd8:=0/18$
cd9:=0/18$
cd10:=-4/18$
cd11:=1/18$
cd12:=-4/18$
cd13:=19/18$
cd14:=-4/18$
cd15:=-2/18$
cd16:=-7/18$
cd17:=1/18$
cd18:=2/18$

% Here are the matrix forms of the elements of $\cD$ corresponding
% to the above sequences of coefficients:

AD:=ad1*ZMAT^0*SIGMAT^0+ad2*ZMAT^0*SIGMAT^1+ad3*ZMAT^0*SIGMAT^2+
ad4*ZMAT^1*SIGMAT^0+ad5*ZMAT^1*SIGMAT^1+ad6*ZMAT^1*SIGMAT^2+
ad7*ZMAT^2*SIGMAT^0+ad8*ZMAT^2*SIGMAT^1+ad9*ZMAT^2*SIGMAT^2+
ad10*i*ZMAT^0*SIGMAT^0+ad11*i*ZMAT^0*SIGMAT^1+ad12*i*ZMAT^0*SIGMAT^2+
ad13*i*ZMAT^1*SIGMAT^0+ad14*i*ZMAT^1*SIGMAT^1+ad15*i*ZMAT^1*SIGMAT^2+
ad16*i*ZMAT^2*SIGMAT^0+ad17*i*ZMAT^2*SIGMAT^1+ad18*i*ZMAT^2*SIGMAT^2$

BD:=bd1*ZMAT^0*SIGMAT^0+bd2*ZMAT^0*SIGMAT^1+bd3*ZMAT^0*SIGMAT^2+
bd4*ZMAT^1*SIGMAT^0+bd5*ZMAT^1*SIGMAT^1+bd6*ZMAT^1*SIGMAT^2+
bd7*ZMAT^2*SIGMAT^0+bd8*ZMAT^2*SIGMAT^1+bd9*ZMAT^2*SIGMAT^2+
bd10*i*ZMAT^0*SIGMAT^0+bd11*i*ZMAT^0*SIGMAT^1+bd12*i*ZMAT^0*SIGMAT^2+
bd13*i*ZMAT^1*SIGMAT^0+bd14*i*ZMAT^1*SIGMAT^1+bd15*i*ZMAT^1*SIGMAT^2+
bd16*i*ZMAT^2*SIGMAT^0+bd17*i*ZMAT^2*SIGMAT^1+bd18*i*ZMAT^2*SIGMAT^2$

CD:=cd1*ZMAT^0*SIGMAT^0+cd2*ZMAT^0*SIGMAT^1+cd3*ZMAT^0*SIGMAT^2+
cd4*ZMAT^1*SIGMAT^0+cd5*ZMAT^1*SIGMAT^1+cd6*ZMAT^1*SIGMAT^2+
cd7*ZMAT^2*SIGMAT^0+cd8*ZMAT^2*SIGMAT^1+cd9*ZMAT^2*SIGMAT^2+
cd10*i*ZMAT^0*SIGMAT^0+cd11*i*ZMAT^0*SIGMAT^1+cd12*i*ZMAT^0*SIGMAT^2+
cd13*i*ZMAT^1*SIGMAT^0+cd14*i*ZMAT^1*SIGMAT^1+cd15*i*ZMAT^1*SIGMAT^2+
cd16*i*ZMAT^2*SIGMAT^0+cd17*i*ZMAT^2*SIGMAT^1+cd18*i*ZMAT^2*SIGMAT^2$

% Explicitly:
AD:=mat(
((-2*i*z^2+5*i*z+7*i+3*z^2-6*z+3)/18,(i*z^2+11*i*z-32*i-15*z^2+45*z+6)/36,(-i*z^2+4*i*z-i+3)/9),
((5*(i*z^2-4*i*z+i+3*z+3))/18,(2*i*z^2-5*i*z+2*i+3)/9,(-4*i*z^2+i*z+2*i+3*z)/18),
((5*(7*i*z^2-13*i*z-26*i+15*z^2-51*z+12))/36,(5*(i*z^2-4*i*z+i-3*z+9))/18,(-2*i*z^2+5*i*z+7*i-3*z^2+6*z+9)/18))$

BD:=mat(
((3*i*z-21*i+5*z^2-14*z-7)/18,(3*i*z^2-9*i*z+6*i+5*z^2-17*z-4)/36,(9*i*z^2-12*i*z-9*i-2*z^2+11*z+7)/90),
((-3*i*z^2-3*i*z+18*i-11*z^2+29*z+16)/18,(3*i*z^2-6*i*z-21*i-4*z^2+13*z-7)/18,(3*i*z-2*z^2+5*z-2)/18),
((5*(-3*i*z^2+3*i*z+12*i-z^2+7*z-10))/36,(-6*i*z^2+15*i*z+9*i+13*z^2-40*z+13)/18,(-3*i*z^2+3*i*z-12*i-z^2+z-4)/18))$

CD:=mat(
((-7*i*z^2+19*i*z-4*i+3*z^2+3*z-24)/18,(i*z^2-4*i*z+i-3*z+9)/18,(i*(z^2-z-2))/9),
((5*(-i*z^2+i*z+2*i-3*z^2+9*z))/18,(11*i*z^2-35*i*z-4*i+3*z^2-9*z-12)/18,(-i*z^2+4*i*z-i+3)/9),
((5*(i*z^2-4*i*z+i+3*z+3))/18,(5*(-i*z^2+i*z+2*i+3*z^2-9*z))/18,(-2*i*z^2+8*i*z-5*i-3*z^2+3*z)/9))$

ADSTAR:=sub(i=-i,tp(AD))$
BDSTAR:=sub(i=-i,tp(BD))$
CDSTAR:=sub(i=-i,tp(CD))$

% Checking that the elements $\xi$ satisfy $\iota(\xi)\xi=1$:
% The following are zero:
ADSTAR*FF*AD-FF;
BDSTAR*FF*BD-FF;
CDSTAR*FF*CD-FF;

% Checking the integrality conditions:

condmatq2type2:=mat(
(1,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0,0),
(0,1,0,0,5,0,0,1,0,0,0,0,0,0,0,0,0,0),
(0,0,1,0,0,5,0,0,1,0,0,1,0,0,5,0,0,1),
(0,0,5,0,0,3,0,0,0,0,0,0,0,0,1,0,0,6),
(1,0,0,7,0,0,0,0,0,0,0,0,5,0,0,6,0,0),
(0,1,0,0,2,0,0,2,0,0,1,0,0,4,0,0,6,0),
(0,5/2,0,0,1,0,0,5,0,0,11/2,0,0,6,0,0,5,0),
(0,0,5/2,0,0,1,0,0,5,0,0,11/2,0,0,6,0,0,5),
(1,0,0,7,0,0,0,0,0,0,0,0,3,0,0,2,0,0),
(0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0,0),
(0,0,0,0,0,0,0,0,0,0,1,0,0,5,0,0,1,0),
(0,0,7,0,0,3,0,0,7,0,0,1,0,0,5,0,0,1),
(0,0,0,0,0,7,0,0,2,0,0,5,0,0,3,0,0,0),
(0,0,0,3,0,0,2,0,0,1,0,0,7,0,0,0,0,0),
(0,7,0,0,4,0,0,2,0,0,1,0,0,2,0,0,2,0),
(0,5/2,0,0,2,0,0,3,0,0,5/2,0,0,1,0,0,5,0),
(0,0,5/2,0,0,2,0,0,3,0,0,5/2,0,0,1,0,0,5),
(0,0,0,5,0,0,6,0,0,1,0,0,7,0,0,0,0,0))$


advec:=mat((ad1),(ad2),(ad3),(ad4),(ad5),(ad6),(ad7),(ad8),(ad9),(ad10),(ad11),(ad12),(ad13),(ad14),(ad15),(ad16),(ad17),(ad18))$

bdvec:=mat((bd1),(bd2),(bd3),(bd4),(bd5),(bd6),(bd7),(bd8),(bd9),(bd10),(bd11),(bd12),(bd13),(bd14),(bd15),(bd16),(bd17),(bd18))$

cdvec:=mat((cd1),(cd2),(cd3),(cd4),(cd5),(cd6),(cd7),(cd8),(cd9),(cd10),(cd11),(cd12),(cd13),(cd14),(cd15),(cd16),(cd17),(cd18))$

% The following are zero, checking that condmatq2type1*advec, condmatq2type1*bdvec and condmatq2type1*cdvec
% have entries in $\Z_2$:
condmatq2type2*advec -  (1/9)*mat((-12),(54),(21),(13),(-13),(21),(-30),(21),(-14),(15),(6),(39),(7),(15),(46),(-1),(4),(15));
condmatq2type2*bdvec - (1/45)*mat((-180),(-105),(0),(55),(-225),(-50),(-10),(-29),(-240),(-15),(-45),(4),(-4),(-80),(-115),(-10),(22),(-100));
condmatq2type2*cdvec -  (1/9)*mat((-3),(-3),(-6),(5),(25),(-3),(3),(-12),(20),(42),(-9),(-6),(-13),(72),(23),(10),(-1),(81));

% The following are zero, checking that condmatq3type1*advec, condmatq3type1*bdvec and condmatq3type1*cdvec
% have entries in $\Z_3$:
condmatq3type1*advec - (1/4)*mat((-5),(5),(25),(37),(18),(36),(-7),(14),(15),(44),(-11),(8),(23),(37),(20),(19),(16),(1));
condmatq3type1*bdvec - (1/20)*mat((-137),(-11),(-3),(-73),(-200),(-44),(33),(28),(-109),(-80),(7),(-26),(-37),(-43),(-70),(11),(-46),(-5));
condmatq3type1*cdvec - (1/2)*mat((1),(8),(0),(-3),(-4),(6),(3),(-1),(2),(14),(1),(0),(-2),(14),(-4),(2),(-3),(6));

% The following are zero, checking the statements about determinants:
det(AD)-1;
det(BD)-(3+4*i)/5;
det(CD)-1;

% Checking that the relations in the group presentation are indeed satisfied:
gendt:=(3+4*i)/5$
gendti:=(3-4*i)/5$
ADI:=AD^(-1)$
CDI:=CD^(-1)$
BDI:=gendti*ADI*BD*ADI*BD*ADI$ % To get a form of BD^(-1) with integer denominators.

% The following are zero:

(ADI*BD)^3 - gendt*ID3;
(CD*BDI*CD)^3 - gendti*ID3;
(ADI*BDI*AD*BDI)^3 - gendti^2*ID3;;
ADI*CDI*BD*AD*BD*CDI*BD*CD*BDI*CD*BD*CDI^2 - gendt*ID3;
ADI*CDI*BD*AD*BD*ADI^2*BD*ADI^2*BDI*CD*BDI^2 - ID3;
BDI*ADI*CDI*BD*CDI*BD*CDI*AD*BDI^2*AD^2*BD*ADI*CD - ID3;
CD^2*BDI*CD*AD^2*BDI*AD^2*BDI*AD*CDI*BD*AD^2*BDI*AD - gendti*ID3;
CDI^2*BD*CDI*AD*BD*ADI*BD^2*ADI*BDI*ADI*CDI*BD*AD^2*BDI*AD - gendt*ID3;
CDI*BD*AD*BD*AD*BDI*CD^2*BDI*CDI*BD*CDI*AD*BD*AD*BD*ADI*CDI*ADI - gendt*ID3;
CDI*BD*CDI*BD*CDI*BDI*ADI*CDI*BD*CDI*AD*BDI^2*AD*CDI*BD*AD^2*BDI - ID3;
ADI*CD*BD*CDI*AD*BDI*ADI^3*BDI*ADI*BDI*CD*BD*ADI^2*BDI*CD*BDI - gendti*ID3;
CD*BDI*CD*AD*BDI*ADI^2*BD^2*ADI*BDI*ADI*CDI*BDI*ADI*CDI*BD*AD*BD*ADI - ID3;
BDI*ADI*CDI*BD*AD^2*BDI*CD*BDI*CD*BDI*CD*AD*CD*BDI*CD*AD*BDI*ADI*BDI - gendti^2*ID3;
CDI*BD*CDI^2*AD*BDI*ADI*CDI*BD*AD*BD*ADI*BD*CD*BDI*CD*AD*BDI*ADI*BDI*ADI - ID3;
CD*BD*CDI*AD*BDI*ADI^3*CDI*BD*CD*BDI*CD^2*AD*BD*ADI*BD^2*ADI^2 - gendt*ID3;
BDI*ADI*BD*ADI^2*BDI*CD*BDI*ADI*BD*CDI*BD*CDI^2*AD^2*BD*ADI*CD*BDI*CDI - ID3;
BDI*CD*AD*BD*ADI*CDI*BD*CD*BDI*CD*BDI^2*AD*BDI*ADI*BDI*CD*BDI*ADI*CD^2*BDI*CD - gendti^2*ID3;
BDI*ADI*CDI*BD*ADI*BD^2*ADI*CD*BDI*CD*AD*BD*AD*CDI^2*BD*CDI*BDI*AD*BDI*ADI*BDI - ID3;
AD^2*BD*ADI*CD*BDI*CDI*BD*AD*BD*ADI*BD^2*ADI^2*CD*BDI*CD*AD*BDI*ADI^2*BD - gendt*ID3;
CD*BDI*CD*BDI*CD*AD*CDI*BD*CDI*BDI*CD*AD*CDI*AD*BDI*ADI*BDI*ADI*CDI*BD*CDI^2*AD - gendti*ID3;

% Looking at the intersection of the two groups $\bar\Gamma_{(a=1,p=5,\emptyset)}$ and $\bar\Gamma_{(a=1,p=5,\{2\})}$.
% The following are zero. The three words in AA,BB and CC generate in <AA,BB,CC>
% its intersection with <AD,BD,CD>, as do the three words in AD,BD and CD. 
AA - CD;
BB*CC*BB*AA*CC*BB*CC - gendt*BDI*ADI*CDI*BD*AD;
AAI*CC*BBI*AA*BB*AA*BB*AA -  BDI*ADI*CDI*BD*AD*BD*AD;


;end;