/*
U3(5):2 presented on its standard generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^4,
(x*y)^10,
(x*y*x*y^-1*x*y^2)^2,
// (x*y^2)^12, // Redundant, and probably useless.
(x,y*x*y)^4,
(x*y*x*y*x*y*x*y^2)^7
>;
c:=x;d:=y;

K1:=sub<G|x*y>;
// C10.

M1:=sub<G|y,x*y*x>;
M2:=sub<G|x,y^-1*x*y*x*y*x*y>;
M3:=sub<G|x,y*x*y*x*y^2*x*y^2>;
M4:=sub<G|x,y^2*x*y*x*y^2>;
M5:=sub<G|x,(x*y*x*y^2*x*y)^2>;
M6:=sub<G|x,(y*x*y)^4>;
// Maximal subgroups: U3(5), S7, 5^{1+2}:8:2, A6.2^2, 2.S5.2, L2(7):2.

/*
Demonstration of correctness:
Coset enumeration over K = C10. Index is 25200.
The following should work:

10*Index(G,K1:Hard:=true,Print:=2,CosetLimit:=10^5);
Index(G,M1:Hard:=true,Print:=2,CosetLimit:=10^4);
Index(G,M2:Hard:=true,Print:=2,CosetLimit:=10^4);
Index(G,M3:Hard:=true,Print:=2,CosetLimit:=10^4);
Index(G,M4:Hard:=true,Print:=2,CosetLimit:=10^4);
Index(G,M5:Hard:=true,Print:=2,CosetLimit:=10^4);
Index(G,M6:Hard:=true,Print:=2,CosetLimit:=10^4);
*/