/*
M20 presented on its standard generators.
*/

G<x,y>:=Group<x,y|
x^4,
y^3,
(x*y)^5,
(x*y^-1)^5,
(x^2*y)^3,
(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2
>;

// Subgroups in `proper' order.
H5:=sub<G|y,x*y*x^2*y^-1*x,x*y*x^-1*y*x*y^-1*x>;
H6:=sub<G|y,x*y*x*y^-1*x^-1>;
H6a:=sub<G|x,y*x*y*x*y^-1>;
H7:=sub<G|y,x*y*x*y^-1*x>;
H8:=sub<G|y,x*y^-1*x*y*x*y*x^2>;




// Class list:
C:=[Id(G),x^2,x*y*x*y*x^-1*y*x*y^-1,x,x*y*x^2*y^-1,x*y*x*y*x^-1*y*x^-1*y^-1,
y,x*y,x*y*x*y];
D:=[u^G:u in C];


H1:=sub<G|y,x*y*x*y*x^-1>;
H2:=sub<G|y,x*y*x*y*x^2>;
H3:=sub<G|y,x*y*x^-1*y*x^-1>;
H4:=sub<G|y,x^-1*y*x^-1*y*x>;





G<x,y,z>:=Group<x,y,z|z^2,(x,z),(y,z),x^4,y^3,(x*y)^5,
(x*y^-1)^5*z,(x^2*y)^3*z,(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2>;

H1:=sub<G|y,x*y*x*y*x^-1*z>;
H2:=sub<G|y,x*y*x*y*x^2*z>;
H3:=sub<G|y,x*y*x^-1*y*x^-1>;
H4:=sub<G|y,x^-1*y*x^-1*y*x>;


F<i>:=QuadraticField(-1);
G<x,y>:=MatrixGroup<4,F|\[-1,0,0,0,0,0,1,0,0,0,0,1,0,-1,-1,-1],
[0,1,0,0,(1+i)/2,(1-i)/2,(1+i)/2,0,(1-i)/2,(-1+i)/2,(-1+i)/2,0,(-1+i)/2,(1-i)/2,(1-i)/2,1]>;
z:=(x*y^-1)^5;

// 4b"M20.
G<x,y,z>:=Group<x,y,z|z^4,(x,z),(y,z),y^3,(x*y)^5,
(x*y^-1)^5*z^-1,(x^2*y)^3*z^-1,
(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2*z^2>;
H1:=sub<G|y,x*y*x*y*x^-1*z^-1>;
H2:=sub<G|y,x*y*x*y*x^2*z>;
// H3:=sub<G|y,x*y*x^-1*y*x^-1>;
// H4:=sub<G|y,x^-1*y*x^-1*y*x>;

> for i in [#S..1 by -1] do H:=sub<G|S[i],z>;A:=MatrixAlgebra<F,1|[[1]:u in Generators(S[i])] cat [[5]]>;HM:=GModule(H,A);GM:=Induction(HM,G);\
CO:=Constituents(GM);print i,"\t",#S[i],"\t",Dimension(GM),"\t",[Dimension(u):u in CO];end for;
32       48      20      [ 4, 16 ]
31       60      16      [ 16 ]
30       60      16      [ 16 ]
29       24      40      [ 16, 24 ]
28       24      40      [ 4, 16, 20 ]


// 4a"M20.
G<x,y,z>:=Group<x,y,z|z^4,(x,z),(y,z),y^3,(x*y)^5,
(x*y^-1)^5*z^-1,(x^2*y)^3*z,
(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2*z^2>;
H1:=sub<G|y,x*y*x*y*x^-1*z^-1>;
H2:=sub<G|y,x*y*x*y*x^2*z>;
// H3:=sub<G|y,x*y*x^-1*y*x^-1>;
// H4:=sub<G|y,x^-1*y*x^-1*y*x>;
27       60      16      [ 16 ]
26       60      16      [ 16 ]
25       24      40      [ 16, 24 ]
24       24      40      [ 16, 24 ]
23       12      80      [ 16, 24 ]
22       12      80      [ 16, 24 ]
21       12      80      [ 16, 24 ]
20       12      80      [ 16, 24 ]
19       8       120     [ 8, 8, 16, 24 ]
18       8       120     [ 8, 8, 16, 24 ]
17       8       120     [ 8, 8, 16, 24 ]
16       10      96      [ 8, 8, 16, 24 ]
15       10      96      [ 8, 8, 16, 24 ]
14       6       160     [ 8, 8, 16, 24 ]
13       6       160     [ 8, 8, 16, 24 ]


for i in [0..3],j in [0..3],k in [0..3] do
G<x,y,z,Z>:=Group<x,y,z,Z|Z^4,z^4,(x,Z),(y,Z),(x,z),(y,z),x^4*Z^2,y^3,
(x*y)^5,(x*y^-1)^5*z^-1*Z^i,(x^2*y)^3*z^-1*Z^j,
(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2*z^2*Z^k>;
if #G eq 15360 then
print i,j,k,"\t",#G,"\t",AQInvariants(G);end if;end for;

for i in [0..3],j in [0..3],k in [0..3] do
G<x,y,z,Z>:=Group<x,y,z,Z|Z^4,z^4,(x,Z),(y,Z),(x,z),(y,z),x^4,y^3,
(x*y)^5,(x*y^-1)^5*z^-1*Z^i,(x^2*y)^3*z^-1*Z^j,
(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2*z^2*Z^k>;
if #G eq 15360 then
print i,j,k,"\t",#G,"\t",AQInvariants(G);end if;end for;


G<x,y,z,Z>:=Group<x,y,z,Z|Z^4,z^4,(x,Z),(y,Z),(x,z),(y,z),x^4*Z^2,y^3,
(x*y)^5,(x*y^-1)^5*z^-1,(x^2*y)^3*z^-1*Z,
(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2*z^2>;

H1:=sub<G|y,x*y*x*y*x^-1*z>;
H2:=sub<G|y,x*y*x*y*x^2*z*Z^2>;
H3:=sub<G|y,x*y*x^-1*y*x^-1>;
H4:=sub<G|y,x^-1*y*x^-1*y*x>;




for j in [1,-1],k in [1,-1] do
G<x,y,z>:=Group<x,y,z|z^4,(x,z),(y,z),y^3,(x*y)^5,
(x*y^-1)^5*z^j,(x^2*y)^3*z^k,
(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2*z^2>;
H1:=sub<G|y,x*y*x*y*x^-1*z^j>;
H2:=sub<G|y,x*y*x*y*x^2*z^-j>;
H3:=sub<G|y,x*y*x^-1*y*x^-1>;
H4:=sub<G|y,x^-1*y*x^-1*y*x>;
aq:={AQInvariants(u):u in [H1,H2,H3,H4]};
print j,k,"\t",#G,"\t",#H1,#H2,#H3,#H4,"\t",#aq,"\t",Random(aq);end for;

for j in [1,3],k in [1,3],l in [0,2] do
G<x,y,z>:=Group<x,y,z|z^4,(x,z),(y,z),y^3,(x*y)^5,
(x*y^-1)^5*z^j,(x^2*y)^3*z^k,
(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2*z^l>;
print j,k,l,"\t",#G,"\t",#G/Index(G,sub<G|x*y*x*y*x^-1*y*x*y^-1>),"\t",AQInvariants(G);end for;


for i in [0,1],j in [0,1],k in [0,1],l in [0,1] do
G<x,y,z,zz>:=Group<x,y,z,zz|zz^2*z,z^2,(x,z),(y,z),(x,zz),(y,zz),x^4*z^i,y^3,
(x*y)^5,(x*y^-1)^5=zz*z^j,(x^2*y)^3=zz*z^k,
(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2=z^l>;
print i,j,k,l,"\t",#G,"\t",AQInvariants(G);end for;

for i in [0,1],j in [0,1],k in [0,1],l in [0,1] do
G<x,y,z,zz>:=Group<x,y,z,zz|zz^2*z,z^2,(x,z),(y,z),(x,zz),(y,zz),x^4*z^i,y^3,
(x*y)^5,(x*y^-1)^5=zz*z^j,(x^2*y)^3=zz*z^k,
(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2=zz*z^l>;
print i,j,k,l,"\t",#G,"\t",AQInvariants(G);end for;


for i in [0,1],j in [0,1],k in [0,1],l in [0,1] do
G<x,y,z,zz>:=Group<x,y,z,zz|zz^2*z,z^2,(x,z),(y,z),(x,zz),(y,zz),x^4*z^i,y^3,
(x*y)^5,(x*y^-1)^5=z^j,(x^2*y)^3=z^k,
(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2=zz*z^l>;
print i,j,k,l,"\t",#G,"\t",AQInvariants(G);end for;



G<x,y,z>:=Group<x,y,z|z^2,(x,z),(y,z),x^4,y^3,(x*y)^5,
(x*y^-1)^5*z,(x^2*y)^3*z,(x*y*x*y^-1*x*y^-1*x^-1*y^-1)^2*z>;
H:=sub<G|(x^2),(x^2)^(y^-1),(x^2)^(y^-1*x),(x^2)^(y^-1*x*y)>;