let h^2=6;
let z^6=-z^3-1$ % 9-th root of 1

phiz:=z^4$
phi2z:=z^7$

s:=1+2*z^3$ % square root of -3.

U:=h*s/3$   % square root of -2.

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

ZEROMAT:=mat(
(0,0,0),
(0,0,0),
(0,0,0))$

ZMAT:=mat((z,  0,  0),
          (0,z^4,  0),
          (0,  0,z^7))$

gendt:=(h+s)/3$  
gendti:=(h-s)/3$ 

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

SIGI:=mat(
(0,0,gendti),
(1,0,0),
(0,1,0))$

T:=3+h*(-1-z^4-z^5)$
phiT:=3 + h*(-1+z-z^2+z^4)$
phi2T:=3 + h*(-1-z+z^2+z^5)$

TI:=(3*(-2*z^5-2*z^2+2*z-1)+(-z^5+z^4-2*z^2+2*z-3)*h)/9$
phiTI:=(3*(2*z^5+2*z^4-1)+(2*z^5+z^4+z^2-z-3)*h)/9$
phi2TI:=(3*(-2*z^4+2*z^2-2*z-1)+(-z^5-2*z^4+z^2-z-3)*h)/9$

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

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

% Generators of group ${\bar\Gamma}(C_{18},p=3,\emptyset)$ are AA, BB and ZZ:

AA:=(1/18)*mat(
((-h*s+(3*h+6))*z^2+(-h*s+(3*h+6))*z+(2*h+3)*s-3,
((3*h+9)*s+(3*h-9))*z^2+((-h-12)*s-3*h-12)*z+(4*h+6)*s+6*h+12,
((-2*h-6)*s+(6*h+12))*z^2+((5*h+12)*s-3*h-12)*z+(-4*h-9)*s-3),
(((-3*h-9)*s-3*h-9)*z^2+(-6*h-12)*s*z+(-3*h-9)*s+3*h+9,
((-h-3)*s-3*h-3)*z^2+((2*h+3)*s-3)*z+(2*h+3)*s-3,
(-3*h*s+(3*h+18))*z^2+(-h*s+(3*h+24))*z+(4*h+6)*s+6*h+12),
(((-5*h-3)*s-3*h-3)*z^2+(4*h*s-6*h-12)*z+(4*h+12)*s+6,
((3*h+9)*s-3*h-9)*z^2+((3*h+6)*s+(9*h+18))*z+(-3*h-9)*s+3*h+9,
((2*h+3)*s-3)*z^2+((-h-3)*s-3*h-3)*z+(2*h+3)*s-3))$

BB:=(1/18)*mat(
(((2*h+3)*s+3)*z+(h+3)*s-3*h-3,
((-2*h-6)*s+6)*z^2+((h+9)*s-3*h-3)*z+(-2*h-3)*s-3,
((h+9)*s-3*h+3)*z^2+((-5*h-3)*s-3*h+3)*z+(h+9)*s+3*h+3),
(((-h+6)*s+(3*h+12))*z^2+((4*h+3)*s+15)*z+(4*h+3)*s-3,
(-h*s-3*h-6)*z+(h+3)*s-3*h-3,
(h*s-3*h-12)*z^2+((-2*h-6)*s-12)*z+(-2*h-3)*s-3),
(((3*h+4)*s+h)*z^2+(7*s+(4*h+9))*z+(-h-5)*s+h+3,
((-h-9)*s-3*h+3)*z^2+((-2*h+6)*s-6*h-12)*z+(4*h+3)*s-3,
((-h-3)*s+(3*h+3))*z+(h+3)*s-3*h-3))$

ZZ:=mat(
(z,        0,         0),
(0,z*(s-1)/2,         0),
(0,        0,-z*(s+1)/2))$

% Generators of group {\bar\Gamma}(C_{18},p=3,\{2\}) are AD, BD and CD:

AD:=(1/18)*mat(
(((h+3)*s-3*h-3)*z^2-h*s-3*h-6,
(h*s+(3*h+6))*z^2+(h*s-3*h-12)*z+(h-3)*s+3*h+15,
((-2*h-6)*s+12)*z^2+((h+9)*s-3*h+3)*z+(-5*h-3)*s-3*h+3),
(((-2*h-3)*s-6*h-3)*z^2+((-h-9)*s-3*h+3)*z+(-2*h-9)*s+6*h-3,
(h*s+(3*h+6))*z^2-h*s-3*h-6,
((-2*h-3)*s-3)*z^2+((-2*h-6)*s+6)*z+(h-3)*s+3*h+15),
((s-2*h-9)*z^2+((3*h+4)*s+h)*z+7*s+4*h+9,
((4*h+3)*s-3)*z^2+((-h+6)*s+(3*h+12))*z+(-2*h-9)*s+6*h-3,
((-2*h-3)*s-3)*z^2-h*s-3*h-6))$

BD:=(1/36)*mat(
((-2*h*s+12)*z^2+(h*s+(3*h+12))*z+(4*h+6)*s-6,
((3*h+6)*s-3*h+18)*z^2+((-2*h+6)*s-12*h-6)*z+(-h+12)*s+9*h+24,
((3*h+6)*s+(3*h-18))*z^2+((-h-12)*s+(3*h-12))*z+(5*h+6)*s+3*h+6),
((3*h*s+9*h)*z^2+(12*s+6*h)*z+(3*h+12)*s-3*h,
((h-6)*s-3*h-6)*z^2+((h+6)*s-3*h-6)*z+(4*h+6)*s-6,
(-12*s+6*h)*z^2+((-5*h-6)*s+(9*h-6))*z+(-h+12)*s+9*h+24),
(((h-12)*s-3*h-12)*z^2+((h+18)*s-3*h+6)*z+(7*h+6)*s+9*h+6,
-6*h*s*z^2+((3*h-6)*s-3*h-18)*z+(3*h+12)*s-3*h,
((h+6)*s+(3*h-6))*z^2+((-2*h-6)*s-6)*z+(4*h+6)*s-6))$

CD:=(1/36)*mat(
(((-3*h-6)*s-3*h-18)*z^2+6*h*z-6*s+18,
((4*h+18)*s-6*h-6)*z^2+((-5*h-6)*s+(3*h-6))*z+(4*h+12)*s+6*h+12,
((3*h+12)*s+9*h)*z^2+((-5*h-12)*s-3*h-24)*z+2*h*s+12*h+24),
(((-2*h-6)*s+(12*h+6))*z^2+((h+12)*s+(9*h+24))*z+(4*h+12)*s+6*h+24,
((3*h+12)*s-3*h)*z^2+(3*h*s-3*h)*z-6*s+18,
((h-6)*s+(9*h+30))*z^2+(4*h*s+(6*h+12))*z+(4*h+12)*s+6*h+12),
(((-5*h-18)*s-3*h+6)*z^2+(-h*s-3*h-24)*z+(6*h+12)*s,
(-5*h*s-9*h-12)*z^2+((4*h+6)*s-6*h-30)*z+(4*h+12)*s+6*h+24,
(-6*s+(6*h+18))*z^2+(-3*h*s-3*h)*z-6*s+18))$

% Here are the inverses of these generators:

AAI:=(1/18)*mat(
(((-2*h-3)*s-3)*z^2+((-2*h-3)*s-3)*z+(-2*h-3)*s-3,
((3*h+6)*s-3*h)*z+(-3*h-3)*s+3*h+9,
((2*h-6)*s+6)*z^2+(2*h*s+6)*z-h*s-3*h+6),
(((h-6)*s-3*h+6)*z^2+(-3*s-9)*z-h*s+3*h-6,
((h+3)*s-3*h-3)*z^2+(h*s+(3*h+6))*z+(-2*h-3)*s-3,
((-3*h-3)*s-3*h-9)*z+(-3*h-3)*s+3*h+9),
(((h+3)*s+(3*h+15))*z-3*s+6*h+9,
(h*s+(3*h-12))*z^2+(-3*s+9)*z-h*s+3*h-6,
(h*s+(3*h+6))*z^2+((h+3)*s-3*h-3)*z+(-2*h-3)*s-3))$

BBI:=(1/18)*mat(
((h*s-3*h-6)*z^2+(-h-3)*s-3*h-3,
(-6*s-6*h)*z^2+((-h+3)*s+(3*h+15))*z+3*s-6*h-9,
((-3*h-1)*s+(7*h+9))*z^2+((2*h+8)*s-2*h-12)*z+(-2*h-7)*s+4*h+3),
(((-4*h-6)*s-12)*z^2+((-4*h-6)*s+6)*z+(-h-3)*s+3*h+15,
((h+3)*s+(3*h+3))*z^2+(-h-3)*s-3*h-3,
((3*h+3)*s+(3*h-9))*z^2+((2*h+6)*s-12)*z+3*s-6*h-9),
(((2*h-3)*s+(6*h+3))*z^2+((-4*h-3)*s+3)*z+(-h-3)*s-3*h-15,
((2*h+9)*s-6*h-3)*z^2+((2*h+6)*s+(6*h+6))*z+(-h-3)*s+3*h+15,
((-2*h-3)*s+3)*z^2+(-h-3)*s-3*h-3))$

ZZI:=mat(
(-z^2*(s+1)/2,          0,  0),
(           0,z^2*(s-1)/2,  0),
(           0,          0,z^2))$

ADI:=(1/18)*mat(
(((2*h+3)*s-3)*z+h*s-3*h-6,
((2*h+6)*s-12)*z^2+((-3*h-6)*s+3*h)*z+(3*h+3)*s+3*h-9,
((2*h+8)*s-2*h-12)*z^2+((-2*h-7)*s+(4*h+3))*z+(5*h+5)*s+h-3),
(((2*h+6)*s+(6*h+6))*z^2+((2*h+9)*s-3)*z+(2*h+9)*s-6*h-3,
((-h-3)*s-3*h-3)*z+h*s-3*h-6,
((-h+3)*s+(3*h+15))*z^2+((3*h+3)*s+(3*h+9))*z+(3*h+3)*s+3*h-9),
(((-4*h-3)*s+3)*z^2+((-h-3)*s-3*h-15)*z+(2*h+3)*s-6*h+3,
((-4*h-6)*s+6)*z^2+((-h-6)*s-3*h-12)*z+(2*h+9)*s-6*h-3,
(-h*s+(3*h+6))*z+h*s-3*h-6))$

BDI:=(1/36)*mat(
(((-h-6)*s-3*h-6)*z^2+((-h-6)*s+(3*h-6))*z+(-4*h-6)*s-6,
((-9*h-6)*s+(3*h+18))*z^2+((3*h+24)*s-3*h)*z+(-9*h-6)*s+3*h-18,
((8*h+18)*s+(6*h+6))*z^2+((-4*h-12)*s-6*h-12)*z+(5*h+12)*s+9*h+24),
(((-h-6)*s+(15*h+30))*z^2+((3*h+6)*s+(9*h+18))*z+(7*h+18)*s+3*h+6,
((2*h+6)*s-6)*z^2+(2*h*s+12)*z+(-4*h-6)*s-6,
((3*h-6)*s-15*h-18)*z^2+((-3*h-12)*s-3*h-36)*z+(-9*h-6)*s+3*h-18),
(((8*h+12)*s-6*h+12)*z^2+((-2*h+12)*s+(12*h+24))*z+(-h-24)*s+3*h+12,
((-7*h-12)*s-9*h-24)*z^2+((3*h+6)*s-9*h-18)*z+(7*h+18)*s+3*h+6,
(-h*s+(3*h+12))*z^2+((-h+6)*s-3*h-6)*z+(-4*h-6)*s-6))$

CDI:=(1/36)*mat(
((-3*h*s-3*h)*z^2+(6*s+(6*h+18))*z+6*s+18,
((2*h+6)*s+6)*z^2+((-7*h-6)*s-3*h+6)*z+(2*h+12)*s-12,
((h+12)*s+(3*h+12))*z^2+(-h*s-3*h-12)*z+6*h),
((-4*h*s+(6*h+12))*z^2+(-h*s+(3*h+12))*z+2*h*s+12,
(3*h*s-3*h)*z^2+((3*h+6)*s-3*h-18)*z+6*s+18,
((-h-6)*s+(3*h+6))*z^2+((2*h+6)*s+(12*h+6))*z+(2*h+12)*s-12),
(((-h-6)*s-3*h-6)*z^2+((-3*h+6)*s-9*h-18)*z+4*h*s-6*h-12,
((-h-6)*s-9*h-6)*z^2+((2*h+6)*s-6)*z+2*h*s+12,
6*h*z^2+((-3*h-12)*s-3*h)*z+6*s+18))$

% Checking that these elements are in the division algebra:

a1:=0$
a2:=0$
a3:=1$
a4:=-2/3$
a5:=1/3$
a6:=-4/3$
a7:=1$
a8:=1/3$
a9:=0$
a10:=-1$
a11:=1/3$
a12:=2/3$
a13:=2/3$
a14:=0$
a15:=-4/3$
a16:=0$
a17:=0$
a18:=1$
a19:=-2/3$
a20:=1/3$
a21:=1/3$
a22:=1$
a23:=-1/3$
a24:=0$
a25:=-1/3$
a26:=-1/3$
a27:=1/3$
a28:=-1/3$
a29:=0$
a30:=-2/3$
a31:=1$
a32:=-1/3$
a33:=1/3$
a34:=-2/3$
a35:=-1/3$
a36:=-1/3$
b1:=0$
b2:=0$
b3:=-1/3$
b4:=-2/3$
b5:=1/3$
b6:=1/3$
b7:=-2/3$
b8:=0$
b9:=0$
b10:=1/3$
b11:=1/3$
b12:=-1/3$
b13:=-1$
b14:=1/3$
b15:=1$
b16:=1/3$
b17:=0$
b18:=-2/3$
b19:=2/3$
b20:=1/3$
b21:=-1/3$
b22:=-2/3$
b23:=1/3$
b24:=1/3$
b25:=1/3$
b26:=0$
b27:=-1/3$
b28:=0$
b29:=1/3$
b30:=0$
b31:=-2/3$
b32:=0$
b33:=1/3$
b34:=0$
b35:=0$
b36:=0$
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$
ad1:=-2/3$
ad2:=-1/3$
ad3:=2/3$
ad4:=-2/3$
ad5:=0$
ad6:=-2/3$
ad7:=1/3$
ad8:=0$
ad9:=1/3$
ad10:=-1$
ad11:=0$
ad12:=-1/3$
ad13:=1/3$
ad14:=0$
ad15:=0$
ad16:=0$
ad17:=1/3$
ad18:=0$
ad19:=-2/3$
ad20:=0$
ad21:=0$
ad22:=1/3$
ad23:=0$
ad24:=1/3$
ad25:=-2/3$
ad26:=1/3$
ad27:=0$
ad28:=-2/3$
ad29:=1/3$
ad30:=-1/3$
ad31:=0$
ad32:=0$
ad33:=1/3$
ad34:=-2/3$
ad35:=1/3$
ad36:=-1/3$
bd1:=1/3$
bd2:=0$
bd3:=1$
bd4:=1/3$
bd5:=1/3$
bd6:=0$
bd7:=0$
bd8:=1/3$
bd9:=2/3$
bd10:=2/3$
bd11:=1/3$
bd12:=2/3$
bd13:=-1/3$
bd14:=0$
bd15:=1/3$
bd16:=0$
bd17:=0$
bd18:=1/3$
bd19:=1/3$
bd20:=1/3$
bd21:=-1/3$
bd22:=-1/6$
bd23:=0$
bd24:=1/6$
bd25:=1/2$
bd26:=-1/6$
bd27:=1/3$
bd28:=1/6$
bd29:=0$
bd30:=-1/2$
bd31:=1/6$
bd32:=-1/6$
bd33:=2/3$
bd34:=1/2$
bd35:=0$
bd36:=1/6$
cd1:=1$
cd2:=1/3$
cd3:=2/3$
cd4:=-1/3$
cd5:=0$
cd6:=-1/3$
cd7:=1/3$
cd8:=-2/3$
cd9:=1/3$
cd10:=2/3$
cd11:=-1/3$
cd12:=2/3$
cd13:=-1$
cd14:=0$
cd15:=-1/3$
cd16:=1/3$
cd17:=-1/3$
cd18:=1$
cd19:=1/6$
cd20:=0$
cd21:=1/6$
cd22:=-1/3$
cd23:=-1/6$
cd24:=-1/2$
cd25:=2/3$
cd26:=-1/6$
cd27:=1/2$
cd28:=-1/3$
cd29:=0$
cd30:=-1/3$
cd31:=1/6$
cd32:=-1/3$
cd33:=-1/6$
cd34:=1/6$
cd35:=1/6$
cd36:=1/3$

% The following are zero, verifying that the given numbers are the coefficient sequences of the 
% various generators:

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

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

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

AD-(((ad1+ad19*U)*ID3+(ad4+ad22*U)*ZZ+(ad7+ad25*U)*ZZ^2+(ad10+ad28*U)*ZZ^3+(ad13+ad31*U)*ZZ^4+(ad16+ad34*U)*ZZ^5)*SIGI+
((ad2+ad20*U)*ID3+(ad5+ad23*U)*ZZ+(ad8+ad26*U)*ZZ^2+(ad11+ad29*U)*ZZ^3+(ad14+ad32*U)*ZZ^4+(ad17+ad35*U)*ZZ^5)*ID3+
((ad3+ad21*U)*ID3+(ad6+ad24*U)*ZZ+(ad9+ad27*U)*ZZ^2+(ad12+ad30*U)*ZZ^3+(ad15+ad33*U)*ZZ^4+(ad18+ad36*U)*ZZ^5)*SIG);

BD-(((bd1+bd19*U)*ID3+(bd4+bd22*U)*ZZ+(bd7+bd25*U)*ZZ^2+(bd10+bd28*U)*ZZ^3+(bd13+bd31*U)*ZZ^4+(bd16+bd34*U)*ZZ^5)*SIGI+
((bd2+bd20*U)*ID3+(bd5+bd23*U)*ZZ+(bd8+bd26*U)*ZZ^2+(bd11+bd29*U)*ZZ^3+(bd14+bd32*U)*ZZ^4+(bd17+bd35*U)*ZZ^5)*ID3+
((bd3+bd21*U)*ID3+(bd6+bd24*U)*ZZ+(bd9+bd27*U)*ZZ^2+(bd12+bd30*U)*ZZ^3+(bd15+bd33*U)*ZZ^4+(bd18+bd36*U)*ZZ^5)*SIG);

CD-(((cd1+cd19*U)*ID3+(cd4+cd22*U)*ZZ+(cd7+cd25*U)*ZZ^2+(cd10+cd28*U)*ZZ^3+(cd13+cd31*U)*ZZ^4+(cd16+cd34*U)*ZZ^5)*SIGI+
((cd2+cd20*U)*ID3+(cd5+cd23*U)*ZZ+(cd8+cd26*U)*ZZ^2+(cd11+cd29*U)*ZZ^3+(cd14+cd32*U)*ZZ^4+(cd17+cd35*U)*ZZ^5)*ID3+
((cd3+cd21*U)*ID3+(cd6+cd24*U)*ZZ+(cd9+cd27*U)*ZZ^2+(cd12+cd30*U)*ZZ^3+(cd15+cd33*U)*ZZ^4+(cd18+cd36*U)*ZZ^5)*SIG);

AASTAR:=sub(z=z^8,tp(AA))$
BBSTAR:=sub(z=z^8,tp(BB))$
ZZSTAR:=sub(z=z^8,tp(ZZ))$

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

ADSTAR:=sub(z=z^8,tp(AD))$
BDSTAR:=sub(z=z^8,tp(BD))$
CDSTAR:=sub(z=z^8,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;

avec:=mat((a1),(a2),(a3),(a4),(a5),(a6),(a7),(a8),(a9),(a10),(a11),(a12),(a13),(a14),(a15),(a16),(a17),(a18),
          (a19),(a20),(a21),(a22),(a23),(a24),(a25),(a26),(a27),(a28),(a29),(a30),(a31),(a32),(a33),(a34),(a35),(a36))$

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))$


advec:=mat((ad1),(ad2),(ad3),(ad4),(ad5),(ad6),(ad7),(ad8),(ad9),(ad10),(ad11),(ad12),(ad13),(ad14),(ad15),(ad16),(ad17),(ad18),
          (ad19),(ad20),(ad21),(ad22),(ad23),(ad24),(ad25),(ad26),(ad27),(ad28),(ad29),(ad30),(ad31),(ad32),(ad33),(ad34),(ad35),(ad36))$

bdvec:=mat((bd1),(bd2),(bd3),(bd4),(bd5),(bd6),(bd7),(bd8),(bd9),(bd10),(bd11),(bd12),(bd13),(bd14),(bd15),(bd16),(bd17),(bd18),
          (bd19),(bd20),(bd21),(bd22),(bd23),(bd24),(bd25),(bd26),(bd27),(bd28),(bd29),(bd30),(bd31),(bd32),(bd33),(bd34),(bd35),(bd36))$

cdvec:=mat((cd1),(cd2),(cd3),(cd4),(cd5),(cd6),(cd7),(cd8),(cd9),(cd10),(cd11),(cd12),(cd13),(cd14),(cd15),(cd16),(cd17),(cd18),
          (cd19),(cd20),(cd21),(cd22),(cd23),(cd24),(cd25),(cd26),(cd27),(cd28),(cd29),(cd30),(cd31),(cd32),(cd33),(cd34),(cd35),(cd36))$

% The following are zero, verifying that AA, BB and ZZ have coefficients in $\Z[1/2,1/3]\cap\Z_2$:
avec-(1/3)*mat((0),(0),(3),(-2),(1),(-4),(3),(1),(0),(-3),(1),(2),(2),(0),(-4),(0),(0),(3),(-2),(1),(1),(3),(-1),(0),(-1),(-1),(1),(-1),(0),(-2),(3),(-1),(1),(-2),(-1),(-1));
bvec-(1/3)*mat((0),(0),(-1),(-2),(1),(1),(-2),(0),(0),(1),(1),(-1),(-3),(1),(3),(1),(0),(-2),(2),(1),(-1),(-2),(1),(1),(1),(0),(-1),(0),(1),(0),(-2),(0),(1),(0),(0),(0));
zvec-mat((0),(0),(0),(0),(1),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0),(0));

% The following are zero, verifying that AA, BB and ZZ have coefficients in $\Z[1/2,1/3]$:
advec-(1/3)*mat((-2),(-1),(2),(-2),(0),(-2),(1),(0),(1),(-3),(0),(-1),(1),(0),(0),(0),(1),(0),(-2),(0),(0),(1),(0),(1),(-2),(1),(0),(-2),(1),(-1),(0),(0),(1),(-2),(1),(-1));
bdvec-(1/6)*mat((2),(0),(6),(2),(2),(0),(0),(2),(4),(4),(2),(4),(-2),(0),(2),(0),(0),(2),(2),(2),(-2),(-1),(0),(1),(3),(-1),(2),(1),(0),(-3),(1),(-1),(4),(3),(0),(1));
cdvec-(1/6)*mat((6),(2),(4),(-2),(0),(-2),(2),(-4),(2),(4),(-2),(4),(-6),(0),(-2),(2),(-2),(6),(1),(0),(1),(-2),(-1),(-3),(4),(-1),(3),(-2),(0),(-2),(1),(-2),(-1),(1),(1),(2));

CondMtxDM2Type2:=(1/2)*mat(
(6,2,2,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),
(4,0,4,0,0,0,0,0,0,2,2,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0),
(2,0,2,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,6,2,2,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0),
(4,0,4,0,0,0,0,0,0,6,0,6,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,2,2,6,0,0,0,0,0,0),
(0,0,0,4,2,2,2,2,4,0,0,0,6,0,2,2,0,6,0,0,0,0,0,4,0,4,0,0,0,0,4,0,4,0,0,0),
(0,0,0,2,0,6,6,0,2,0,0,0,2,2,4,4,2,2,0,0,0,4,0,4,0,0,0,0,0,0,4,0,0,0,4,0),
(0,0,0,4,0,2,0,2,0,0,0,0,2,0,2,0,0,4,0,0,0,4,2,2,2,2,4,0,0,0,6,0,2,2,0,6),
(0,0,0,6,0,6,0,0,4,0,0,0,6,0,4,0,2,4,0,0,0,2,0,6,6,0,2,0,0,0,2,2,4,4,2,2),
(0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,4,0,4),
(0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,0,4,4),
(0,0,0,0,0,0,2,6,4,0,0,0,0,0,0,2,0,6,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,4),
(0,0,0,0,0,0,6,0,2,0,0,0,0,0,0,4,6,2,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,0,0,4),
(0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,6,6,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,4),
(0,0,0,0,0,0,2,2,6,0,0,0,0,0,0,2,6,6,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,4,0,4),
(0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,2,0,2,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,6,6,2),
(0,0,0,0,0,0,6,0,6,0,0,0,0,0,0,6,0,6,0,0,0,0,0,0,2,2,6,0,0,0,0,0,0,2,6,6),
(4,2,2,2,2,4,0,0,0,6,0,2,2,0,6,0,0,0,0,0,4,0,4,0,0,0,0,4,0,4,0,0,0,0,0,0),
(2,0,6,6,0,2,0,0,0,2,2,4,4,2,2,0,0,0,4,0,4,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0),
(4,0,2,0,2,0,0,0,0,2,0,2,0,0,4,0,0,0,4,2,2,2,2,4,0,0,0,6,0,2,2,0,6,0,0,0),
(6,0,6,0,0,4,0,0,0,6,0,4,0,2,4,0,0,0,2,0,6,6,0,2,0,0,0,2,2,4,4,2,2,0,0,0),
(0,0,0,4,0,0,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,4,0,4,0,0,0),
(0,0,0,4,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,0,4,4,0,0,0),
(0,0,0,2,6,4,0,0,0,0,0,0,2,0,6,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,4,0,0,0),
(0,0,0,6,0,2,0,0,0,0,0,0,4,6,2,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,0,0,4,0,0,0),
(0,0,0,4,0,4,0,0,0,0,0,0,6,0,6,0,6,4,0,0,0,4,0,4,4,0,4,0,0,0,2,2,6,2,6,6),
(0,0,0,2,0,2,0,2,4,0,0,0,2,0,2,0,6,4,0,0,0,6,6,2,6,2,2,0,0,0,6,2,2,6,6,2),
(0,0,0,2,0,6,2,0,6,0,0,0,3,7,5,7,1,5,0,0,0,4,0,4,0,0,0,0,0,0,6,0,6,0,6,4),
(0,0,0,5,1,3,1,7,3,0,0,0,5,7,3,1,1,3,0,0,0,2,0,2,0,2,4,0,0,0,2,0,2,0,6,4),
(0,0,4,2,4,4,6,0,6,0,0,0,2,0,2,6,0,4,2,0,2,2,6,4,2,0,6,0,0,0,2,0,6,2,2,4),
(0,0,0,6,0,6,2,0,4,0,0,4,4,4,6,4,0,2,0,0,0,6,0,2,6,6,4,2,0,2,4,6,2,4,2,2),
(3,0,7,7,1,6,3,0,5,0,0,0,7,0,1,3,7,6,0,0,4,2,4,4,6,0,6,0,0,0,2,0,2,6,0,4),
(0,0,0,1,0,7,5,1,2,3,0,7,6,1,7,6,7,3,0,0,0,6,0,6,2,0,4,0,0,4,4,4,6,4,0,2),
(6,2,4,6,6,4,0,0,0,6,0,2,6,0,2,0,0,0,4,0,0,0,4,0,0,0,0,4,0,4,0,0,0,0,0,0),
(2,0,6,2,0,6,0,0,0,4,2,6,4,6,6,0,0,0,4,0,4,0,0,0,0,0,0,0,0,4,0,4,0,0,0,0),
(2,0,4,0,6,0,0,0,0,2,0,2,0,0,4,0,0,0,6,2,4,6,6,4,0,0,0,6,0,2,6,0,2,0,0,0),
(6,0,6,0,0,4,0,0,0,4,0,6,0,6,4,0,0,0,2,0,6,2,0,6,0,0,0,4,2,6,4,6,6,0,0,0))$

% The following are zero, verifying that the vectors CondMtxDM2Type2*xvec, for xd =ad, bd and cd,
% have entries in $\Z[1/2,1/3]\cap\Z_2$, so that AD, BD and CD fix the type 2 vertex $o_2'$:
CondMtxDM2Type2*advec-(1/3)*mat((-17),(-12),(-20),(-20),(6),(4),(-7),(-10),(-6),(0),(-7),(1),(1),(-3),(-4),(3),(-24),(-15),(-11),(-15),(2),(-2),(-1),(0),(-1),(-2),(6),(2),(4),(-14),(-8),(-6),(-30),(-17),(-9),(-14));
CondMtxDM2Type2*bdvec-(1/3)*mat((14),(15),(13),(16),(14),(6),(14),(22),(8),(12),(17),(9),(9),(19),(15),(16),(13),(19),(21),(21),(6),(7),(10),(6),(18),(20),(19),(15),(31),(21),(35),(25),(21),(19),(21),(25));
CondMtxDM2Type2*cdvec-(1/3)*mat((22),(13),(11),(20),(-6),(-4),(2),(-7),(16),(20),(14),(15),(10),(19),(19),(29),(4),(1),(2),(15),(-13),(-14),(-11),(-17),(-8),(2),(2),(-10),(16),(-2),(15),(-5),(3),(3),(1),(9));

% The following are zero:
det(AA)-1;
det(BB)-gendt;
det(ZZ)-z^3;

% The following are zero:
det(AD)-z^6*gendt;
det(BD)-1;
det(CD)-z^3*gendt;

% The following are zero, verifying that the claimed relations hold:

ZZ^3 - z^3*ID3;
(BB*ZZ)^3 - z^3*gendt*ID3;
AA^2*ZZ*BB*ZZI*AA*ZZI*AA^3*BBI*AA*ZZ - z^3*ID3;
AAI*ZZI*BB*ZZ*AAI*ZZI*BB*ZZ*AAI*BB - z^3*gendt*ID3;
BB*AAI*BB*AAI^2*ZZ*BBI^2*ZZ*AA*ZZI*AA*ZZI - z^6*ID3;
ZZ*AA*ZZ*AA^2*BBI*AA*ZZ*AAI*ZZ*AAI*ZZI*BB - ID3;
BB*ZZ*AAI*BBI*ZZ*AAI*ZZ*BBI*ZZ*AAI*ZZI*BB - ID3;
BBI*AAI*BB*AAI^2*BB*ZZI*AA*ZZI*AA*ZZI*BBI - ID3;
ZZI*AAI*BB*AAI^2*ZZI*BBI^2*ZZ*AAI*BBI*ZZ*AA*BBI - z^3*gendti*ID3;
BB*AAI*ZZI*BB*AA*ZZI*BBI*ZZ*AAI*BB*ZZI*AA*ZZI*BB*AAI - gendt*ID3;
BBI*ZZ*AAI*ZZ*BBI*ZZ*AA*ZZI*BBI^2*ZZ*AA^2*ZZI*BBI^2*ZZ*AA - gendti^2*ID3;

% The following are zero, verifying that the claimed relations hold:

(ADI*CDI)^3 - gendti^2*ID3;
(ADI*BDI*CD*ADI*BD)^3 - gendti*ID3;
(BDI*ADI*CD*ADI*BDI)^3 - gendti*ID3;
CDI*BD*AD*BD*CD*ADI - z^6*ID3;
ADI*BDI*AD*BD^2*AD*CDI*AD*BD*ADI*BD*CDI*ADI*CDI*BD - z^3*gendti*ID3;
ADI*CD*BD*CD*ADI^2*CD*AD*CD*BDI*AD*CDI*BD*AD*BDI - gendt*ID3;
AD*CDI*BD*AD*CDI*AD*BD^2*CD*ADI^2*BD*CDI*ADI*CDI - z^3*gendti*ID3;
AD*CDI*BD*AD*CDI^2*ADI*BDI*CD*ADI*BDI*CD*ADI*CD*AD*BDI - z^3*ID3;
BDI*ADI*CD*BDI*AD^2*CDI*BDI*AD*CDI*AD*BD^3*CDI^2-ID3;
BD^2*CDI*ADI*CDI*BD*ADI*BDI^2*CD*ADI*CDI*ADI*BDI*AD*CDI - gendti^2*ID3;
BDI^2*ADI*BDI*ADI*BDI*AD^2*CDI*AD*CDI*AD*BD^2*AD*CDI - z^3*ID3;
BDI*CD^2*BDI^3*ADI*CD^2*BDI*CD*BDI*CD*ADI*BD*ADI*BDI - gendt*ID3;
CDI*AD*CDI*BD*AD*BDI*ADI*CD*ADI*BDI*CD*ADI*CDI*BDI*ADI*CD*ADI - z^3*gendti*ID3;
BDI^3*ADI*CD*ADI*BDI*AD*CDI*BDI*CD^2*BDI^3*ADI*CD*ADI - z^3*ID3;
CDI*ADI*BDI*AD^2*CDI*BD*CDI^2*ADI*BDI*CD^2*ADI^2*CDI*ADI*BDI - z^3*gendti^2*ID3;
BD*CD*ADI^2*CDI*ADI*BDI*CD*BDI*ADI*BD*CDI^2*BDI*ADI^2*BDI*CD - z^3*gendti^2*ID3;
BD*CD*ADI^2*BD*ADI*CDI^2*ADI*BDI*CD*BDI*ADI*BD*CDI*ADI*CDI*BD*CD^2 - z^6*gendti^2*ID3;
BD*CDI^2*ADI*BDI*CD*BDI*ADI*BD*AD*BDI*ADI*CDI*BDI*CD*AD*CD*BDI*AD*BD - z^3*ID3;
AD^2*CDI*BDI*CDI*AD^2*CDI*BD*CDI*AD^2*CDI*BDI*AD*CDI*BDI^2*ADI*BDI - z^6*ID3;
BDI*CDI*BD*CD*ADI*BDI*ADI*BD*AD*CD*BDI*ADI*CDI^2*AD*BD^3*AD*CD - z^6*ID3;
AD*BD*ADI*BDI*AD*BD*AD*CDI*BDI*CD^2*BDI^2*ADI*BD*AD*CD*AD*BD*CD - gendt^2*ID3;
CDI*AD*BD*CDI*BD*AD*CD^2*BDI*AD*CDI*BDI*CD^2*AD*BDI*ADI*CDI*ADI*BDI*AD - z^3*gendt*ID3;
ADI*BDI*CD*ADI*BD*CDI^2*BDI*ADI*CDI*BDI*ADI*CDI*BDI*ADI*CDI*ADI*BDI^2*CD*BD - z^6*gendti^3*ID3;

;end;


% The following are zero, verifying that CDI*AD*CD, CDI*BD*CD and CDI*(CD*BD*CD)*CD 
% are words in the generators of  ${\bar\Gamma}(C_{18},p=3,\emptyset)$:
CDI*AD*CD-z^6*(ZZ^2*AAI*BB*ZZ*AA^2*BBI*AA*BB*ZZ*AA*ZZ);
CDI*BD*CD-z^6*(ZZ*AA*ZZ^2*BBI*AAI*BB*AAI*AAI);
CDI*(CD*BD*CD)*CD-gendt*ZZ^2*BBI*ZZ*AA*ZZ^2*AAI*BB*AAI^2*ZZ*BBI*ZZ^2;