let S^2=-15$
let Z^3=3*Z+3$

phiZ:=(4*s+(3*s-5)*Z-2*s*Z^2)/10$
phi2Z:=(-4*s-(3*s+5)*Z+2*s*Z^2)/10$

off nat;

gendt:=(s+1)/4$
gendti:=(-s+1)/4$

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

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

SIG:=mat((0,1,0),
         (0,0,1),
         (2,0,0))$

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

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

phi2ZMAT:=mat((phi2Z,0,   0),
              (    0,Z,   0),
              (    0,0,phiZ))$

FF:=mat((2,0,0),
        (0,0,1),
        (0,1,0))$

% Inverse of FF:

FFI:=mat((1/2,0,0),
         (  0,0,1),
         (  0,1,0))$

% Here are the generators of $\bar\Gamma$ in the case $(a=15,p=2,\emptyset)$:

AE:=(1/30)*mat(
(2*s*z^2-3*s*z-9*s+5*z-5,-3*s*z^2+2*s*z+6*s-5*z^2+10*z+20,10*(z^2-z-2)),
(2*(-s*z^2-s*z+2*s-5*z^2+5*z+10),-2*s*z^2+3*s*z-s+5*z-5,s*z^2+s*z-2*s-5*z^2-5*z+20),
(2*(2*s*z^2-3*s*z-4*s+10*z^2-5*z-10),2*(s*z^2+s*z-2*s-5*z^2+5*z+10),5*(-s-2*z-1)))$

BE:=(1/60)*mat(
(-10*(s+1),2*(3*s*z^2-2*s*z-6*s-5*z^2+20),s*z^2+s*z-2*s+5*z^2-15*z-10),
(2*(s*z^2-4*s*z-2*s+5*z^2-10),-10*(s+1),20*(z^2-1)),
(4*(-3*s*z^2+2*s*z+6*s-5*z^2+20),2*(-2*s*z^2+3*s*z+4*s-10*z^2+15*z+20),-10*(s+1)))$

% Here are the generators of $\bar\Gamma$ in the case $(a=15,p=2,\{3\})$:

A3:=(1/30)*mat(
(2*s*z^2-3*s*z-9*s+15*z-15,0,0),
(0,-4*s*z^2+6*s*z+3*s-15,0),
(0,0,2*s*z^2-3*s*z-9*s-15*z-15))$

B3:=(1/12)*mat(
(2*(s-3),2*(-s-3),-s+3),
(2*(-s+3),2*(s-3),2*(-s-3)),
(4*(-s-3),2*(-s+3),2*(s-3)))$

% Here are the generators of $\bar\Gamma$ in the case $(a=15,p=2,\{5\})$:

A5:=(1/30)*mat(
(-2*s*z^2+3*s*z-s-5*z-5,2*s*z^2-s*z-5*s-8*z^2+17*z+11,2*s*z^2-3*s*z-4*s+10*z^2-5*z-20),
(2*(-3*s*z^2+2*s*z+6*s-5*z^2+10*z+10),2*s*z^2-3*s*z-9*s-5*z-5,-2*s*z^2+4*s*z+3*s+10*z^2-10*z-25),
(2*(-3*s*z-s-2*z^2-7*z-1),2*(s*z^2+s*z-2*s-5*z^2-5*z+10),5*(-s+2*z-1)))$

B5:=(1/60)*mat(
(-10*(s+1),2*(3*s*z^2-2*s*z-6*s-5*z^2+20),s*z^2+s*z-2*s+5*z^2-15*z-10),
(2*(s*z^2-4*s*z-2*s+5*z^2-10),-10*(s+1),20*(z^2-1)),
(4*(-3*s*z^2+2*s*z+6*s-5*z^2+20),2*(-2*s*z^2+3*s*z+4*s-10*z^2+15*z+20),-10*(s+1)))$

% Here are the generators of $\bar\Gamma$ in the case $(a=15,p=2,\{3,5\})$:

A35:=(1/30)*mat(
(2*s*z^2-3*s*z-9*s+15*z-15,0,0),
(0,-4*s*z^2+6*s*z+3*s-15,0),
(0,0,2*s*z^2-3*s*z-9*s-15*z-15))$

B35:=(1/60)*mat(
(2*(2*s*z^2-3*s*z-4*s+10*z^2-15*z),2*(3*s*z+s+8*z^2+3*z-21),5*(-s+4*z^2-9)),
(2*(-6*s*z^2+4*s*z+7*s-10*z^2+15),2*(-s*z^2-s*z+2*s-5*z^2+15*z+30),2*(-3*s*z^2+s*z+7*s+5*z^2-15*z-15)),
(4*(3*s*z^2-4*s*z-5*s-13*z^2+12*z+21),2*(6*s*z^2-4*s*z-17*s-10*z^2+15),2*(-s*z^2+4*s*z+2*s-5*z^2+30)))$

% Here are the inverses of these generators:

AEI:=(1/30)*mat(
(-2*s*z^2+3*s*z+9*s+5*z-5,-2*s*z^2+3*s*z+4*s+10*z^2-5*z-10,s*z^2+s*z-2*s-5*z^2+5*z+10),
(20*(z^2-z-2),5*(s-2*z-1),-s*z^2-s*z+2*s-5*z^2-5*z+20),
(2*(3*s*z^2-2*s*z-6*s-5*z^2+10*z+20),2*(-s*z^2-s*z+2*s-5*z^2+5*z+10),2*s*z^2-3*s*z+s+5*z-5))$

BEI:=(1/60)*mat(
(10*(s-1),2*(3*s*z^2-2*s*z-6*s-5*z^2+20),-s*z^2+4*s*z+2*s+5*z^2-10),
(2*(-s*z^2-s*z+2*s+5*z^2-15*z-10),10*(s-1),20*(z^2-1)),
(4*(-3*s*z^2+2*s*z+6*s-5*z^2+20),2*(2*s*z^2-3*s*z-4*s-10*z^2+15*z+20),10*(s-1)))$

A3I:=(1/30)*mat(
(-2*s*z^2+3*s*z+9*s+15*z-15,0,0),
(0,-2*s*z^2+3*s*z+9*s-15*z-15,0),
(0,0,4*s*z^2-6*s*z-3*s-15))$

B3I:=(1/12)*mat(
(2*(-s-3),2*(s-3),s+3),
(2*(s+3),2*(-s-3),2*(s-3)),
(4*(s-3),2*(s+3),2*(-s-3)))$

A5I:=(1/30)*mat(
(2*s*z^2-3*s*z+s-5*z-5,3*s*z+s-2*z^2-7*z-1,3*s*z^2-2*s*z-6*s-5*z^2+10*z+10),
(2*(-2*s*z^2+3*s*z+4*s+10*z^2-5*z-20),5*(s+2*z-1),2*s*z^2-4*s*z-3*s+10*z^2-10*z-25),
(2*(-2*s*z^2+s*z+5*s-8*z^2+17*z+11),2*(-s*z^2-s*z+2*s-5*z^2-5*z+10),-2*s*z^2+3*s*z+9*s-5*z-5))$

B5I:=(1/60)*mat(
(10*(s-1),2*(3*s*z^2-2*s*z-6*s-5*z^2+20),-s*z^2+4*s*z+2*s+5*z^2-10),
(2*(-s*z^2-s*z+2*s+5*z^2-15*z-10),10*(s-1),20*(z^2-1)),
(4*(-3*s*z^2+2*s*z+6*s-5*z^2+20),2*(2*s*z^2-3*s*z-4*s-10*z^2+15*z+20),10*(s-1)))$

A35I:=(1/30)*mat(
(-2*s*z^2+3*s*z+9*s+15*z-15,0,0),
(0,-2*s*z^2+3*s*z+9*s-15*z-15,0),
(0,0,4*s*z^2-6*s*z-3*s-15))$

B35I:=(1/60)*mat(
(2*(-2*s*z^2+3*s*z+4*s+10*z^2-15*z),2*(-3*s*z^2+4*s*z+5*s-13*z^2+12*z+21),6*s*z^2-4*s*z-7*s-10*z^2+15),
(10*(s+4*z^2-9),2*(s*z^2-4*s*z-2*s-5*z^2+30),2*(3*s*z^2-s*z-7*s+5*z^2-15*z-15)),
(4*(-3*s*z-s+8*z^2+3*z-21),2*(-6*s*z^2+4*s*z+17*s-10*z^2+15),2*(s*z^2+s*z-2*s-5*z^2+15*z+30)))$

ae1:=-1/6$ ae2:=2/3$ ae3:=-2/3$ ae4:=1/6$ ae5:=1/3$ ae6:=-1/3$ ae7:=0$ ae8:=-1/6$ ae9:=1/3$ ae10:=-3/10$ ae11:=1/5$ ae12:=0$ ae13:=-1/10$ ae14:=1/15$ ae15:=0$ ae16:=1/15$ ae17:=-1/10$ ae18:=0$ 

be1:=-1/6$ be2:=2/3$ be3:=-1/6$ be4:=0$ be5:=0$ be6:=-1/4$ be7:=0$ be8:=-1/6$ be9:=1/12$ be10:=-1/6$ be11:=-1/5$ be12:=-1/30$ be13:=0$ be14:=-1/15$ be15:=1/60$ be16:=0$ be17:=1/10$ be18:=1/60$ 

a3_1:=-1/2$ a3_2:=0$ a3_3:=0$ a3_4:=1/2$ a3_5:=0$ a3_6:=0$ a3_7:=0$ a3_8:=0$ a3_9:=0$ a3_10:=-3/10$ a3_11:=0$ a3_12:=0$ a3_13:=-1/10$ a3_14:=0$ a3_15:=0$ a3_16:=1/15$ a3_17:=0$ a3_18:=0$ 

b3_1:=-1/2$ b3_2:=-1/2$ b3_3:=1/4$ b3_4:=0$ b3_5:=0$ b3_6:=0$ b3_7:=0$ b3_8:=0$ b3_9:=0$ b3_10:=1/6$ b3_11:=-1/6$ b3_12:=-1/12$ b3_13:=0$ b3_14:=0$ b3_15:=0$ b3_16:=0$ b3_17:=0$ b3_18:=0$ 

a5_1:=-1/6$ a5_2:=11/30$ a5_3:=-2/3$ a5_4:=-1/6$ a5_5:=17/30$ a5_6:=-1/6$ a5_7:=0$ a5_8:=-4/15$ a5_9:=1/3$ a5_10:=-1/30$ a5_11:=-1/6$ a5_12:=-2/15$ a5_13:=1/10$ a5_14:=-1/30$ a5_15:=-1/10$ a5_16:=-1/15$ a5_17:=1/15$ a5_18:=1/15$ 

b5_1:=-1/6$ b5_2:=2/3$ b5_3:=-1/6$ b5_4:=0$ b5_5:=0$ b5_6:=-1/4$ b5_7:=0$ b5_8:=-1/6$ b5_9:=1/12$ b5_10:=-1/6$ b5_11:=-1/5$ b5_12:=-1/30$ b5_13:=0$ b5_14:=-1/15$ b5_15:=1/60$ b5_16:=0$ b5_17:=1/10$ b5_18:=1/60$ 

a35_1:=-1/2$ a35_2:=0$ a35_3:=0$ a35_4:=1/2$ a35_5:=0$ a35_6:=0$ a35_7:=0$ a35_8:=0$ a35_9:=0$ a35_10:=-3/10$ a35_11:=0$ a35_12:=0$ a35_13:=-1/10$ a35_14:=0$ a35_15:=0$ a35_16:=1/15$ a35_17:=0$ a35_18:=0$ 

b35_1:=0$ b35_2:=-7/10$ b35_3:=-3/4$ b35_4:=-1/2$ b35_5:=1/10$ b35_6:=0$ b35_7:=1/3$ b35_8:=4/15$ b35_9:=1/3$ b35_10:=-2/15$ b35_11:=1/30$ b35_12:=-1/12$ b35_13:=-1/10$ b35_14:=1/10$ b35_15:=0$ b35_16:=1/15$ b35_17:=0$ b35_18:=0$ 

aevec:=mat((ae1),(ae2),(ae3),(ae4),(ae5),(ae6),(ae7),(ae8),(ae9),(ae10),(ae11),(ae12),(ae13),(ae14),(ae15),(ae16),(ae17),(ae18))$

bevec:=mat((be1),(be2),(be3),(be4),(be5),(be6),(be7),(be8),(be9),(be10),(be11),(be12),(be13),(be14),(be15),(be16),(be17),(be18))$

a3vec:=mat((a3_1),(a3_2),(a3_3),(a3_4),(a3_5),(a3_6),(a3_7),(a3_8),(a3_9),(a3_10),(a3_11),(a3_12),(a3_13),(a3_14),(a3_15),(a3_16),(a3_17),(a3_18))$

b3vec:=mat((b3_1),(b3_2),(b3_3),(b3_4),(b3_5),(b3_6),(b3_7),(b3_8),(b3_9),(b3_10),(b3_11),(b3_12),(b3_13),(b3_14),(b3_15),(b3_16),(b3_17),(b3_18))$

a5vec:=mat((a5_1),(a5_2),(a5_3),(a5_4),(a5_5),(a5_6),(a5_7),(a5_8),(a5_9),(a5_10),(a5_11),(a5_12),(a5_13),(a5_14),(a5_15),(a5_16),(a5_17),(a5_18))$

b5vec:=mat((b5_1),(b5_2),(b5_3),(b5_4),(b5_5),(b5_6),(b5_7),(b5_8),(b5_9),(b5_10),(b5_11),(b5_12),(b5_13),(b5_14),(b5_15),(b5_16),(b5_17),(b5_18))$

a35vec:=mat((a35_1),(a35_2),(a35_3),(a35_4),(a35_5),(a35_6),(a35_7),(a35_8),(a35_9),(a35_10),(a35_11),(a35_12),(a35_13),(a35_14),(a35_15),(a35_16),(a35_17),(a35_18))$

b35vec:=mat((b35_1),(b35_2),(b35_3),(b35_4),(b35_5),(b35_6),(b35_7),(b35_8),(b35_9),(b35_10),(b35_11),(b35_12),(b35_13),(b35_14),(b35_15),(b35_16),(b35_17),(b35_18))$

CondMtxDM3Type1:=mat(
(1,2,4,0,6,6,0,6,3,0,0,0,0,3,3,0,0,0),
(0,0,3,1,8,4,0,6,6,0,6,0,0,0,0,0,6,6),
(0,8,5,0,0,3,1,8,1,0,0,0,0,3,0,0,0,0),
(0,6,6,0,6,3,3,6,3,0,3,3,0,0,0,0,0,0),
(1,8,4,0,6,6,3,6,3,0,0,0,0,6,6,0,0,0),
(0,0,3,1,8,1,0,6,6,0,3,0,0,0,0,0,0,0),
(0,6,3,3,6,3,0,0,0,0,0,0,0,0,0,0,0,0),
(0,6,6,3,6,3,3,6,3,0,6,6,0,0,0,0,0,0),
(1,8,1,0,6,6,3,6,3,0,0,0,0,0,0,0,0,0),
(0,6,0,0,1,7,0,0,0,1,2,4,0,6,6,0,6,3),
(0,5,0,0,3,3,0,5,2,0,0,3,1,8,4,0,6,6),
(0,6,0,0,4,0,0,0,6,0,8,5,0,0,3,1,8,1),
(0,1,7,0,0,0,0,3,3,0,6,6,0,6,3,3,6,3),
(0,3,3,0,5,2,0,6,0,1,8,4,0,6,6,3,6,3),
(0,4,0,0,0,6,0,3,0,0,0,3,1,8,1,0,6,6),
(0,0,0,0,3,3,0,3,3,0,6,3,3,6,3,0,0,0),
(0,5,2,0,6,0,0,6,6,0,6,6,3,6,3,3,6,3),
(0,0,6,0,3,0,0,3,0,1,8,1,0,6,6,3,6,3))$

CondMtxDM3Type2:=mat(
(1,2,4,0,6,6,0,6,3,0,0,0,0,3,3,0,0,0),
(0,0,3,1,8,4,0,6,6,0,6,0,0,0,0,0,6,6),
(0,0,0,0,3,0,0,0,0,0,6,6,0,0,0,3,6,3),
(0,6,6,0,6,3,3,6,3,0,3,3,0,0,0,0,0,0),
(1,8,4,0,6,6,3,6,3,0,0,0,0,6,6,0,0,0),
(0,3,0,0,0,0,0,0,0,0,0,0,3,6,3,0,0,0),
(0,0,0,0,3,3,0,3,3,0,6,3,3,6,3,0,0,0),
(0,5,2,0,6,0,0,6,6,0,6,6,3,6,3,3,6,3),
(1,8,1,0,6,6,3,6,3,0,0,0,0,0,0,0,0,0),
(0,6,0,0,1,7,0,0,0,1,2,4,0,6,6,0,6,3),
(0,5,0,0,3,3,0,5,2,0,0,3,1,8,4,0,6,6),
(0,8,5,0,0,3,1,8,1,0,0,0,0,3,0,0,0,0),
(0,1,7,0,0,0,0,3,3,0,6,6,0,6,3,3,6,3),
(0,3,3,0,5,2,0,6,0,1,8,4,0,6,6,3,6,3),
(0,0,3,1,8,1,0,6,6,0,3,0,0,0,0,0,0,0),
(0,5,4,7,2,7,0,3,0,0,0,0,0,3,3,0,3,3),
(0,2,2,7,2,4,7,8,7,0,5,2,0,6,0,0,6,6),
(0,0,6,0,3,0,0,3,0,1,8,1,0,6,6,3,6,3))$

CondMtxDM5Type1:=mat(
(1,0,0,18,0,0,24,0,0,0,0,0,0,0,0,0,0,0),
(0,1,0,0,18,0,0,24,0,0,0,0,0,0,0,0,0,0),
(0,0,1,0,0,18,0,0,24,0,0,0,0,0,0,0,0,0),
(0,0,2,0,0,7,0,0,7,0,0,0,0,0,20,0,0,15),
(1,0,0,16,0,0,16,0,0,0,0,0,10,0,0,20,0,0),
(0,1,0,0,16,0,0,16,0,0,0,0,0,10,0,0,20,0),
(0,2,0,0,7,0,0,7,0,0,0,0,0,5,0,0,10,0),
(0,0,2,0,0,7,0,0,7,0,0,0,0,0,5,0,0,10),
(1,0,0,16,0,0,16,0,0,0,0,0,15,0,0,5,0,0),
(0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,24,0,0),
(0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,24,0),
(0,0,0,0,0,0,0,0,0,0,0,1,0,0,18,0,0,24),
(0,0,0,0,0,12,0,0,9,0,0,2,0,0,7,0,0,7),
(0,0,0,6,0,0,17,0,0,1,0,0,16,0,0,16,0,0),
(0,0,0,0,6,0,0,17,0,0,1,0,0,16,0,0,16,0),
(0,0,0,0,13,0,0,16,0,0,2,0,0,7,0,0,7,0),
(0,0,0,0,0,13,0,0,16,0,0,2,0,0,7,0,0,7),
(0,0,0,19,0,0,8,0,0,1,0,0,16,0,0,16,0,0))$

CondMtxDM5Type2:=(1/5)*mat(
(5,0,0,90,0,0,120,0,0,0,0,0,0,0,0,0,0,0),
(0,5,0,0,90,0,0,120,0,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,0,0,0,0,50,0,0,25,0,0,75),
(0,0,10,0,0,35,0,0,35,0,0,0,0,0,100,0,0,75),
(5,0,0,80,0,0,80,0,0,0,0,0,50,0,0,100,0,0),
(0,0,0,0,50,0,0,100,0,0,50,0,0,50,0,0,50,0),
(0,0,0,0,65,0,0,80,0,0,10,0,0,35,0,0,35,0),
(0,0,0,0,0,65,0,0,80,0,0,10,0,0,35,0,0,35),
(5,0,0,80,0,0,80,0,0,0,0,0,75,0,0,25,0,0),
(0,0,0,0,0,0,0,0,0,5,0,0,90,0,0,120,0,0),
(0,0,0,0,0,0,0,0,0,0,5,0,0,90,0,0,120,0),
(0,0,5,0,0,90,0,0,120,0,0,0,0,0,0,0,0,0),
(0,0,0,0,0,60,0,0,45,0,0,10,0,0,35,0,0,35),
(0,0,0,30,0,0,85,0,0,5,0,0,80,0,0,80,0,0),
(0,5,0,0,80,0,0,80,0,0,0,0,0,50,0,0,100,0),
(0,41,0,0,56,0,0,31,0,0,0,0,0,65,0,0,80,0),
(0,0,41,0,0,56,0,0,31,0,0,0,0,0,65,0,0,80),
(0,0,0,95,0,0,40,0,0,5,0,0,80,0,0,80,0,0))$

% The following are zero:

AE-(ae1*ZMAT^0*SIG^0+ae2*ZMAT^0*SIG^1+ae3*ZMAT^0*SIG^2+
ae4*ZMAT^1*SIG^0+ae5*ZMAT^1*SIG^1+ae6*ZMAT^1*SIG^2+
ae7*ZMAT^2*SIG^0+ae8*ZMAT^2*SIG^1+ae9*ZMAT^2*SIG^2+
ae10*s*ZMAT^0*SIG^0+ae11*s*ZMAT^0*SIG^1+ae12*s*ZMAT^0*SIG^2+
ae13*s*ZMAT^1*SIG^0+ae14*s*ZMAT^1*SIG^1+ae15*s*ZMAT^1*SIG^2+
ae16*s*ZMAT^2*SIG^0+ae17*s*ZMAT^2*SIG^1+ae18*s*ZMAT^2*SIG^2);

BE-(be1*ZMAT^0*SIG^0+be2*ZMAT^0*SIG^1+be3*ZMAT^0*SIG^2+
be4*ZMAT^1*SIG^0+be5*ZMAT^1*SIG^1+be6*ZMAT^1*SIG^2+
be7*ZMAT^2*SIG^0+be8*ZMAT^2*SIG^1+be9*ZMAT^2*SIG^2+
be10*s*ZMAT^0*SIG^0+be11*s*ZMAT^0*SIG^1+be12*s*ZMAT^0*SIG^2+
be13*s*ZMAT^1*SIG^0+be14*s*ZMAT^1*SIG^1+be15*s*ZMAT^1*SIG^2+
be16*s*ZMAT^2*SIG^0+be17*s*ZMAT^2*SIG^1+be18*s*ZMAT^2*SIG^2);

A3-(a3_1*ZMAT^0*SIG^0+a3_2*ZMAT^0*SIG^1+a3_3*ZMAT^0*SIG^2+
a3_4*ZMAT^1*SIG^0+a3_5*ZMAT^1*SIG^1+a3_6*ZMAT^1*SIG^2+
a3_7*ZMAT^2*SIG^0+a3_8*ZMAT^2*SIG^1+a3_9*ZMAT^2*SIG^2+
a3_10*s*ZMAT^0*SIG^0+a3_11*s*ZMAT^0*SIG^1+a3_12*s*ZMAT^0*SIG^2+
a3_13*s*ZMAT^1*SIG^0+a3_14*s*ZMAT^1*SIG^1+a3_15*s*ZMAT^1*SIG^2+
a3_16*s*ZMAT^2*SIG^0+a3_17*s*ZMAT^2*SIG^1+a3_18*s*ZMAT^2*SIG^2);

B3-(b3_1*ZMAT^0*SIG^0+b3_2*ZMAT^0*SIG^1+b3_3*ZMAT^0*SIG^2+
b3_4*ZMAT^1*SIG^0+b3_5*ZMAT^1*SIG^1+b3_6*ZMAT^1*SIG^2+
b3_7*ZMAT^2*SIG^0+b3_8*ZMAT^2*SIG^1+b3_9*ZMAT^2*SIG^2+
b3_10*s*ZMAT^0*SIG^0+b3_11*s*ZMAT^0*SIG^1+b3_12*s*ZMAT^0*SIG^2+
b3_13*s*ZMAT^1*SIG^0+b3_14*s*ZMAT^1*SIG^1+b3_15*s*ZMAT^1*SIG^2+
b3_16*s*ZMAT^2*SIG^0+b3_17*s*ZMAT^2*SIG^1+b3_18*s*ZMAT^2*SIG^2);

A5-(a5_1*ZMAT^0*SIG^0+a5_2*ZMAT^0*SIG^1+a5_3*ZMAT^0*SIG^2+
a5_4*ZMAT^1*SIG^0+a5_5*ZMAT^1*SIG^1+a5_6*ZMAT^1*SIG^2+
a5_7*ZMAT^2*SIG^0+a5_8*ZMAT^2*SIG^1+a5_9*ZMAT^2*SIG^2+
a5_10*s*ZMAT^0*SIG^0+a5_11*s*ZMAT^0*SIG^1+a5_12*s*ZMAT^0*SIG^2+
a5_13*s*ZMAT^1*SIG^0+a5_14*s*ZMAT^1*SIG^1+a5_15*s*ZMAT^1*SIG^2+
a5_16*s*ZMAT^2*SIG^0+a5_17*s*ZMAT^2*SIG^1+a5_18*s*ZMAT^2*SIG^2);

B5-(b5_1*ZMAT^0*SIG^0+b5_2*ZMAT^0*SIG^1+b5_3*ZMAT^0*SIG^2+
b5_4*ZMAT^1*SIG^0+b5_5*ZMAT^1*SIG^1+b5_6*ZMAT^1*SIG^2+
b5_7*ZMAT^2*SIG^0+b5_8*ZMAT^2*SIG^1+b5_9*ZMAT^2*SIG^2+
b5_10*s*ZMAT^0*SIG^0+b5_11*s*ZMAT^0*SIG^1+b5_12*s*ZMAT^0*SIG^2+
b5_13*s*ZMAT^1*SIG^0+b5_14*s*ZMAT^1*SIG^1+b5_15*s*ZMAT^1*SIG^2+
b5_16*s*ZMAT^2*SIG^0+b5_17*s*ZMAT^2*SIG^1+b5_18*s*ZMAT^2*SIG^2);

A35-(a35_1*ZMAT^0*SIG^0+a35_2*ZMAT^0*SIG^1+a35_3*ZMAT^0*SIG^2+
a35_4*ZMAT^1*SIG^0+a35_5*ZMAT^1*SIG^1+a35_6*ZMAT^1*SIG^2+
a35_7*ZMAT^2*SIG^0+a35_8*ZMAT^2*SIG^1+a35_9*ZMAT^2*SIG^2+
a35_10*s*ZMAT^0*SIG^0+a35_11*s*ZMAT^0*SIG^1+a35_12*s*ZMAT^0*SIG^2+
a35_13*s*ZMAT^1*SIG^0+a35_14*s*ZMAT^1*SIG^1+a35_15*s*ZMAT^1*SIG^2+
a35_16*s*ZMAT^2*SIG^0+a35_17*s*ZMAT^2*SIG^1+a35_18*s*ZMAT^2*SIG^2);

B35-(b35_1*ZMAT^0*SIG^0+b35_2*ZMAT^0*SIG^1+b35_3*ZMAT^0*SIG^2+
b35_4*ZMAT^1*SIG^0+b35_5*ZMAT^1*SIG^1+b35_6*ZMAT^1*SIG^2+
b35_7*ZMAT^2*SIG^0+b35_8*ZMAT^2*SIG^1+b35_9*ZMAT^2*SIG^2+
b35_10*s*ZMAT^0*SIG^0+b35_11*s*ZMAT^0*SIG^1+b35_12*s*ZMAT^0*SIG^2+
b35_13*s*ZMAT^1*SIG^0+b35_14*s*ZMAT^1*SIG^1+b35_15*s*ZMAT^1*SIG^2+
b35_16*s*ZMAT^2*SIG^0+b35_17*s*ZMAT^2*SIG^1+b35_18*s*ZMAT^2*SIG^2);

AE*AEI-ID3;
BE*BEI-ID3;

A3*A3I-ID3;
B3*B3I-ID3;

A5*A5I-ID3;
B5*B5I-ID3;

A35*A35I-ID3;
B35*B35I-ID3;

det(AE)-1;
det(BE)-gendt;

det(A3)-1;
det(B3)-gendt;

det(A5)-1;
det(B5)-gendt;

det(A35)-1;
det(B35)-gendt;

AESTAR:=sub(S=-S,tp(AE))$
BESTAR:=sub(S=-S,tp(BE))$

A3STAR:=sub(S=-S,tp(A3))$
B3STAR:=sub(S=-S,tp(B3))$

A5STAR:=sub(S=-S,tp(A5))$
B5STAR:=sub(S=-S,tp(B5))$

A35STAR:=sub(S=-S,tp(A35))$
B35STAR:=sub(S=-S,tp(B35))$

% The following are zero, checking that the elements are unitary with respect to $\iota$:
AESTAR*FF*AE-FF;
BESTAR*FF*BE-FF;

A3STAR*FF*A3-FF;
B3STAR*FF*B3-FF;

A5STAR*FF*A5-FF;
B5STAR*FF*B5-FF;

A35STAR*FF*A35-FF;
B35STAR*FF*B35-FF;

CondMtxDM3Type1*aevec - (1/10)*mat((-13),(11),(2),(16),(29),(21),(35),(27),(45),(19),(30),(82),(-23),(13),(0),(18),(59),(-22));
CondMtxDM5Type1*aevec - (1/6)*mat((17),(16),(8),(-8),(17),(12),(11),(-8),(8),(-3),(-6),(0),(-6),(1),(-7),(11),(6),(14));
CondMtxDM3Type1*bevec - (1/20)*mat((-38),(-50),(46),(16),(39),(-37),(55),(2),(55),(38),(41),(72),(-37),(-31),(15),(-53),(28),(-59));
CondMtxDM5Type1*bevec - (1/12)*mat((-2),(-40),(-32),(-11),(-2),(-8),(10),(-15),(-2),(-2),(12),(8),(-25),(-2),(-30),(-34),(-21),(-2));

CondMtxDM3Type2*a3vec - (1/10)*mat((-5),(5),(2),(0),(-5),(-3),(-3),(-1),(-5),(-3),(-1),(0),(2),(-1),(5),(35),(35),(-1));
CondMtxDM5Type1*a3vec - (1/6)*mat((51),(0),(0),(0),(47),(0),(0),(0),(38),(-3),(0),(0),(0),(13),(0),(0),(0),(52));
CondMtxDM3Type2*b3vec - (1/4)*mat((-2),(-1),(-6),(-9),(-14),(-6),(-5),(-14),(-17),(-14),(-11),(-11),(-1),(-9),(1),(-6),(-6),(1));
CondMtxDM5Type1*b3vec - (1/12)*mat((-6),(-6),(3),(6),(-6),(-6),(-12),(6),(-6),(2),(-2),(-1),(-2),(2),(-2),(-4),(-2),(2));

CondMtxDM3Type1*a5vec - (1/10)*mat((-7),(19),(-28),(-4),(11),(21),(26),(-18),(39),(5),(22),(47),(-60),(-23),(-2),(-2),(27),(-50));
CondMtxDM5Type2*a5vec - (1/6)*mat((-19),(25),(-5),(-7),(-19),(-6),(18),(16),(-10),(1),(5),(26),(3),(-3),(37),(50),(-33),(-16));
CondMtxDM3Type1*b5vec - (1/20)*mat((-38),(-50),(46),(16),(39),(-37),(55),(2),(55),(38),(41),(72),(-37),(-31),(15),(-53),(28),(-59));
CondMtxDM5Type2*b5vec - (1/12)*mat((-2),(-40),(0),(-11),(-2),(-60),(-34),(-21),(-2),(-2),(12),(-32),(-25),(-2),(-8),(62),(-38),(-2));

CondMtxDM3Type2*a35vec - (1/10)*mat((-5),(5),(2),(0),(-5),(-3),(-3),(-1),(-5),(-3),(-1),(0),(2),(-1),(5),(35),(35),(-1));
CondMtxDM5Type2*a35vec - (1/6)*mat((51),(0),(0),(0),(47),(0),(0),(0),(38),(-3),(0),(0),(0),(13),(0),(0),(0),(52));
CondMtxDM3Type2*b35vec - (1/20)*mat((-18),(37),(4),(-93),(-76),(-36),(47),(-12),(-43),(-78),(-15),(-125),(-73),(-33),(35),(-174),(24),(-51));
CondMtxDM5Type2*b35vec - (1/12)*mat((-12),(90),(-10),(10),(-28),(92),(76),(62),(-46),(-4),(22),(87),(34),(24),(74),(-20),(-49),(-90));

% The following are zero, checking that the relations given in the write-up for group $(a=15,p=2,\emptyset)$ really do hold:

(BE*AEI*BE*AE)^3 - gendt^2*ID3;
(BE^2*AEI*BEI*AE*BE)^3 - gendt^2*ID3;
(AEI*BE*AE*BEI*AEI*BEI^2*AEI)^3 - gendti^2*ID3;
BEI*AEI*BE*AE*BEI*AEI*BEI*AE^2*BEI^3*AEI^3*BE*AE*BEI*AEI*BEI - gendti^2*ID3;
AEI^2*BEI*AEI^3*BE*AE*BE*AEI*BEI*AE*BE^3*AE*BEI^3*AEI - ID3;
BE^2*AEI*BEI*AE*BEI*AEI^2*BEI*AE*BEI*AEI*BEI^2*AE^2*BE^3*AE^2*BE - ID3;
BE^2*AE*BE^4*AEI*BE*AE*BE*AEI*BEI*AE^2*BE*AE^3*BE^3*AE*BE - gendt^4*ID3;
AE*BEI^3*AEI^3*BE^2*AE*BE*AE*BEI^3*AEI*BE^2*AE*BE*AEI*BEI*AE^2*BE*AE - ID3;
BE*AEI*BEI*AE*BE^2*AEI*BEI*AE^3*BE^2*AE*BEI*AEI*BEI^2*AEI*BE*AE*BEI*AEI*BEI^2*AE*BE^2 - ID3;
BE*AE*BEI^4*AEI*BE*AEI*BE*AE*BE*AEI*BEI*AEI*BE*AE*BEI^2*AEI^2*BEI*AE*BEI*AEI*BEI^2*AE - gendti^2*ID3;

% The following are zero, checking that the relations given in the write-up for group $(a=15,p=2,\{3\})$ really do hold:

(B3*A3I*B3I*A3*B3*A3)^3 - gendt*ID3;
(B3I*A3I*B3I*A3I*B3^2*A3I^2*B3*A3I)^3 - gendt*ID3;
B3*A3*B3*A3I*B3I*A3*B3*A3^2*B3I*A3*B3*A3^2*B3I*A3*B3*A3^2*B3 - gendt*ID3;
B3I*A3*B3*A3*B3*A3*B3*A3I^2*B3*A3I*B3*A3I^3*B3I*A3I*B3*A3*B3I - gendt*ID3;
A3*B3I*A3^2*B3I^2*A3*B3*A3^2*B3*A3I*B3I*A3I^2*B3I*A3I*B3*A3*B3I - gendti*ID3;
A3*B3I*A3I^2*B3I*A3I*B3*A3I*B3I*A3I*B3I*A3I*B3*A3I*B3I^2*A3I^2*B3I*A3I*B3^2 - gendti*ID3;
A3I*B3*A3^2*B3*A3I^2*B3*A3I*B3*A3I*B3*A3*B3I*A3I*B3*A3*B3I*A3*B3*A3^2*B3*A3I - gendt^2*ID3;
B3*A3^2*B3*A3I*B3I*A3^2*B3I*A3I^2*B3I*A3*B3I^2*A3*B3*A3*B3^2*A3I*B3I*A3*B3*A3^2 - ID3;
B3I*A3I^2*B3I*A3I*B3^2*A3I*B3*A3*B3I*A3*B3^2*A3*B3I*A3*B3*A3^2*B3*A3*B3*A3I^2*B3*A3I*B3*A3I*B3*A3*B3I - gendt^2*ID3;
B3I*A3I*B3*A3*B3I*A3^2*B3I*A3I*B3I^2*A3I*B3*A3*B3I*A3*B3*A3^2*B3*A3I*B3*A3^2*B3I*A3^2*B3I^2*A3*B3*A3 - gendti*ID3;
A3*B3I*A3I*B3I*A3I^2*B3I*A3I*B3^2*A3I*B3*A3^2*B3*A3I*B3*A3*B3*A3I*B3I*A3*B3*A3^2*B3*A3^2*B3*A3I^2*B3 - gendt^2*ID3;
A3^2*B3*A3I^2*B3*A3I*B3*A3I*B3*A3^2*B3*A3I*B3I*A3*B3*A3*B3I*A3*B3^2*A3*B3I*A3*B3*A3^2*B3*A3I*B3I*A3 - gendt^2*ID3;

% The following are zero, checking that the relations given in the write-up for group $(a=15,p=2,\{5\})$ really do hold:

(A5I*B5)^3 - gendt*ID3;
(B5I*A5^3*B5I*A5*B5*A5I*B5I*A5I^3)^3 - gendti^2*ID3;
B5*A5^3*B5I*A5*B5^2*A5^2*B5I*A5I*B5^2*A5*B5I^2*A5*B5I*A5I^2*B5^3 - gendt*ID3;
B5I^2*A5I*B5*A5I^3*B5*A5^2*B5*A5I*B5^3*A5I*B5^3*A5^2*B5*A5*B5I^2*A5 - gendt^2*ID3;
A5^2*B5I*A5*B5*A5I*B5I*A5I^3*B5^2*A5I*B5I*A5I^2*B5I^3*A5^3*B5I*A5^2*B5 - gendti*ID3;
A5*B5I*A5I*B5*A5I^2*B5I*A5I^2*B5I^3*A5^3*B5*A5*B5I^2*A5*B5I*A5I^2*B5^3 - gendti*ID3;
A5^2*B5*A5*B5*A5^2*B5*A5*B5I*A5I*B5*A5I^2*B5I^2*A5^3*B5*A5I*B5^3*A5*B5I^2*A5 - gendt*ID3;
B5*A5*B5*A5I*B5*A5*B5I*A5^4*B5I*A5*B5^2*A5^2*B5I^2*A5*B5I*A5I^2*B5^3*A5^2 - gendt*ID3;
B5*A5*B5I*A5I*B5*A5I^2*B5I*A5I^2*B5I*A5I*B5I*A5I^2*B5I^3*A5^2*B5*A5I*B5*A5^2*B5I*A5*B5 - gendti*ID3;
A5^2*B5I*A5I*B5*A5I^3*B5^2*A5*B5I^2*A5*B5I*A5I*B5I*A5I^3*B5*A5I^2*B5*A5I^3*B5^3 - gendt*ID3;
A5^3*B5I*A5*B5^2*A5^2*B5I*A5^2*B5*A5*B5I^2*A5*B5I*A5I^2*B5*A5^3*B5I*A5*B5*A5I^2*B5 - ID3;
A5*B5I*A5I*B5*A5I*B5I^2*A5*B5I^3*A5*B5I*A5I^2*B5^2*A5^2*B5*A5I*B5^3*A5*B5*A5I*B5*A5*B5*A5I*B5^3 - gendt^2*ID3;

% The following are zero, checking that the relations given in the write-up for group $(a=15,p=2,\{3,5\})$ really do hold:

(B35^3*A35I*B35I*A35)^3 - gendt^2*ID3;
B35I*A35I*B35I*A35I^3*B35I*A35*B35*A35I*B35I*A35*B35^2*A35*B35I^2*A35I - gendti*ID3;
B35I*A35*B35I^2*A35I*B35*A35*B35I*A35^3*B35*A35*B35*A35^2*B35^2*A35*B35I*A35 - ID3;
A35*B35*A35*B35^2*A35I*B35*A35I*B35I^2*A35I*B35*A35*B35I*A35I*B35I*A35I*B35I*A35I^2*B35I*A35*B35 - ID3;
(B35I*A35*B35^2*A35*B35I*A35*B35I)^3 - gendti*ID3;
B35I*A35I*B35*A35*B35I*A35I*B35*A35I*B35I^2*A35I*B35I*A35I^2*B35I*A35I^3*B35I^2*A35I*B35I*A35I*B35*A35 - gendti^2*ID3;
A35^2*B35*A35^2*B35*A35*B35*A35*B35*A35I*B35*A35I*B35*A35I*B35I^2*A35I*B35*A35*B35I^2*A35I*B35*A35*B35I - gendt*ID3;
B35*A35I*B35I^2*A35I^2*B35I*A35*B35*A35*B35^2*A35*B35I*A35*B35*A35I*B35I^2*A35I*B35I*A35I^2*B35I*A35I^3 - gendti*ID3;
A35I*B35I*A35*B35*A35*B35^2*A35*B35I*A35^2*B35*A35^2*B35*A35*B35*A35*B35I*A35I*B35*A35*B35I^3*A35I*B35I*A35I - ID3;
A35*B35*A35^2*B35*A35*B35*A35*B35*A35I*B35I*A35I*B35I*A35I^2*B35I*A35*B35*A35*B35^2*A35*B35I^3*A35I*B35I*A35I*B35*A35*B35I*A35I*B35*A35*B35I - ID3;
B35I*A35I*B35*A35*B35*A35*B35I*A35I*B35*A35*B35I*A35*B35*A35^2*B35*A35*B35*A35*B35*A35I*B35*A35^2*B35*A35^2*B35*A35*B35*A35*B35I*A35I*B35I*A35I - gendt^2*ID3;
A35*B35*A35I*B35I*A35I*B35I*A35I^2*B35I*A35I*B35*A35I*B35I^2*A35I*B35I*A35I*B35*A35*B35*A35*B35I^2*A35I*B35*A35*B35I*A35^2*B35*A35I*B35I*A35*B35*A35I*B35I*A35*B35 - gendti*ID3;
B35^2*A35^3*B35*A35*B35*A35*B35*A35*B35I^2*A35I*B35*A35*B35I*A35*B35*A35I*B35I*A35*B35*A35I*B35I*A35I*B35I*A35I^2*B35I*A35*B35*A35I*B35I*A35*B35*A35*B35*A35 - gendt*ID3;
A35*B35*A35I*B35I*A35I*B35I*A35I^2*B35I*A35*B35*A35^2*B35*A35*B35*A35*B35*A35I*B35^2*A35I*B35I*A35*B35^2*A35I*B35I*A35I*B35I*A35I*B35I*A35I*B35*A35I*B35I*A35*B35*A35I*B35I*A35*B35 - gendt*ID3;
(B35^2*A35*B35I*A35^2*B35*A35I*B35I*A35*B35*A35I*B35I*A35I*B35I*A35*B35*A35)^3 - gendt*ID3;
(B35^2*A35*B35I*A35^2*B35*A35I*B35I*A35*B35*A35I*B35I*A35I*B35I*A35*B35*A35I*B35I*A35*B35*A35)^3 - gendt*ID3;


;end;