/*
2"J2:2 (the non-ATLAS 2.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)^3,
// (x*y*x*y*x*y^-2)^4, // redundant.
(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;

M1a:=sub<G|x,y*x*y^-1,(x,y*x*y)^3*x^(y*x*y)>;
M1b:=sub<G|(x,y*x*y)^3*x,(x,y*x*y)^3*y*x*y^-1,x^(y*x*y)>;
// Reps of both classes of U3(3):2, of index 200.

/*
Proof of correctness (to show that G has most the required order of 2419200)
is probably by coset enumeration over <x,x^y> = D14.
In magma >= 2.8 [and maybe <= 2.7], the following should work.

14*Index(G,sub<G|x,x^y>:Print:=2,Hard:=true,CosetLimit:=7*10^6,Grain:=10^5);
*/