/*
U3(4) presented on its standard generators.
Standard generators of U3(4) are [any] (2,3,13)-generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^3,
(x*y)^13,
(x,y)^5,
// (x,y^-1*(x*y)^4*x*y^-1*x*y*x*y^-1*x*y^-1), // Redundant but useful.
(x,y*x*y*x*y)^3
>;
a:=x;b:=y;

H:=sub<G|x,y*(x,y)^2,y*x*y*x*y*(x,y*x*y*x*y)>;
HH:=sub<G|x,y*(x,y)^2,y^-1*(x*y)^4*x*y^-1*x*y*x*y^-1*x*y^-1>;
// H = HH = C_G(x).

// Maximal subgroups.
M1:=sub<G|y,x*y^-1*x*y*x*y*x*y*x*y^-1*x*y*x*y*x*y^-1*x>;
M2:=sub<G|x,y*x*y^-1*x*y*x*y*x*y^-1*x*y>;
M3:=sub<G|x,y*x*y*x*y^-1>;
M4:=sub<G|y,x*y*x*y^-1*x*y^-1*x*y*x*y*x*y^-1*x*y*x*y*x*y^-1*x>;
M2a:=sub<G|y,x*y*x*y*x*y*x*y^-1*x*y*x*y*x*y*x>;
M3a:=sub<G|y,x*y*x*y^-1*x>;
// 2^{2+4}:15, 5xA5, 5^2:S3, 13:3, 5xA5, 5^2:S3.

/*
Verification of order:
Coset enumeration over <x*y> = 13 or <x,y^2> = D10.

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