/*
J2:2 presented on its standard generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^5,
(x*y)^14,
(x,y)^7,
// (y,x*y*x*y^2*x*y^-1*x*y^2*x*y*x), // redundant but useful.
(x*y*x*y*x*y^-2*x*y^-2)^3,
// (x*y*x*y*x*y^-2)^4, // redundant.
(x,y*x*y)^3, // also redundant, but too good to lose.
(x*y*x*y*x*y^2)^3*x*y^-1*x*y*x*y*x*y^-1*x*y^2
// (x,y*x*y^2)^3 // redundant.
>;

c:=x;d:=y;

H:=sub<G|x,y*(x,y)^3,y*x*y*(x,y*x*y)>;
// H centralises x and has index 1800.

M1:=sub<G|x,y*x*y^-1,x^(y*x*y)>;
// U3(3):2, of index 100.