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

G<x,y>:=Group<x,y|
x^2,
y^4,
(x*y)^12,
(x,y)^5,
(x*y^2)^10,
// (x,y*x*y)^3,  // redundant.
// (x,y^2*x*y)^3,  // redundant.
x*y*x*y*x*y^2*x*y*x*y^2*x*y^-1*x*y^2*x*y*x*y*x*y^-1*x*y^2*x*y^-1*x*y^2
>;
e:=x;f:=y;

H:=sub<G|x,(x*y^2)^5,y*(x,y)^2>;
// H = C_G(x) = 2^{2+4}:5:4.

// Maximal subgroups.
M0:=sub<G|x,y^2,y*x*y>;
M1:=sub<G|y,x*y*x*y^2*x*y^-1*x*y*x>;
M2:=sub<G|y,x*y*x*y*x*y*x*y*x*y*x>;
M3:=sub<G|x,y^2*x*y*x*y*x*y^-1>;
M3a:=sub<G|y,x*y*x*y^2*x*y^2*x*y^-1*x>;
M4:=sub<G|y,x*y^2*x*y^2*x*y^2*x*y*x*y^-1*x*y*x*y^2*x>;
// U3(4):2, 2^{2+4}:15:4, (D10xA5).2, 5^2:(4xS3), 5^2:(4xS3), 13:12.

/*
Verification of order:
Coset enumeration over <x*y> = 12 or <x,y^2> = D20.

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