/*
2^2"Sz(8):3 presented on its standard generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^3,
(x*y)^15,
// (x,y)^13, // Redundant.
(x*y*x*y*x*y^-1)^6,
(x,(x,y*x*y*x*y*x*y^-1*x*y^-1)),
(x*y)^5*x*y^-1*(x*y*x*y*x*y*x*y^-1*x*y)^2*(x*y*x*y^-1)^3*x*y^-1
>;

K:=sub<G|x*y>;
// Is C15.

H:=sub<G|y,(x*y)^5*(x,y*x*y*x*y)^5>;
// Stabiliser for a faithful permutation action. Is 2^3:7:3.


/*
Demonstration of correctness.
Coset enumeration over <xy> = 15 establishes the order.
Can work in degree 2080 image of G on the cosets of H to establish the
structure of G.

The following should work:
15*Index(G,K:Print:=2,Hard:=true,CosetLimit:=10^5,Grain:=10^5);
Index(G,H:Print:=2,Hard:=true,CosetLimit:=10^5,Grain:=10^5);

G<x,y>:=CosetImage(G,H);
NL:=NormalLattice(G);NL;
*/