/*
U3(4):2 presented on its standard generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^3,
(x*y)^8,
(x,y)^13,
(x,y*x*y*x*y*x*y^-1*x*y*x*y)^2,
(x,y^-1*x*y*x*y)^5
>;
c:=x;d:=y;

H:=sub<G|x,y*(x,y)^6,(x,y*x*y*x*y*x*y^-1*x*y*x*y)>;
// H = C_G(x) = 2 x A5.

// Maximal subgroups.
M0:=sub<G|y,y^x>;
M1:=sub<G|x,y*x*y^-1*x*y*x*y^-1*x*y*x*y>;
M2:=sub<G|x,y*x*y^-1*x*y^-1*x*y*x*y*x*y^-1*x*y^-1*x*y>;
M3:=sub<G|x,y*x*y*x*y^-1*x*y*x*y^-1*x*y^-1*x*y*x*y*x*y*x*y^-1>;
M3a:=sub<G|x,y*x*y*x*y^-1*x*y*x*y^-1*x*y*x*y^-1*x*y^-1*x*y^-1*x*y>;
M4:=sub<G|x,y*x*y*x*y^-1*x*y*x*y*x*y>;
// U3(4), 2^{2+4}:(3xD10), D10xA5, 5^2:D12, 5^2:D12, 13:6.

/*
Verification of order:
Coset enumeration over <x*y> = 8 or <x,y^2> = D26.

8*Index(G,sub<G|x*y>:Print:=2,Hard:=true,CosetLimit:=10^5,Grain:=30000);
26*Index(G,sub<G|x,x^y>:Print:=2,Hard:=true,CosetLimit:=10^5,Grain:=30000);
*/