/*
J2 presented on its standard generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^3,
(x*y)^7,
(x,y)^12,
// (x*y*x*y*x*y^-1*x*y*x*y^-1)^6, // Is redundant.
(x*y*x*y*x*y^-1*x*y*x*y^-1*x*y^-1*x*y*x*y*x*y^-1*x*y^-1*x*y*x*y^-1)^3
>;

a:=x;b:=y;

K1:=sub<G|x*y>; // C7.
K2:=sub<G|x,x^y>; // D24.

M1:=sub<G|y,x*y*x*y*x*y^-1*x*y*x>;
// U3(3), of index 100.

/*
Coset enumerations that should work [first 2 verify the order]:
7*Index(G,K1:Print:=2,Hard:=true,CosetLimit:=120000,Grain:=30000);
24*Index(G,K2:Print:=2,Hard:=true,CosetLimit:=50000,Grain:=10000);
Index(G,M1:Print:=2,Hard:=true,CosetLimit:=1000,Grain:=1000);
*/