/*
J3:2 presented on its standard generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^3,
(x*y)^24,
(x,y)^9,
(x*y*(x*y*x*y^-1)^2)^4,
(x*y*x*y*x*y^-1*(x*y*x*y*x*y^-1*x*y^-1)^2)^2,
(x,(y*x)^4*(y^-1*x)^2*y)^2,
(x,y*(x*y^-1)^2*(x*y)^4)^2
>;
c:=x;d:=y;

H:=sub<G|x,y*(x,y)^4,y*x*y*(x*y^-1)^2*x*y*(x*y^-1)^3*x*y*(x*y^-1)^2*x*y*x*y>;
// H = C_G(x).
K:=sub<G|x*y>;
// K = C_{24}.

H:=sub<G|x,y*(x,y)^4,y*x*y*(x*y^-1)^2*x*y*(x*y^-1)^3*x*y*(x*y^-1)^2*x*y*x*y>;
// H = C_G(x).
K:=sub<G|x*y>;
// K = C24.

M1:=sub<G|y,x*y*x>;
// J3.
M2:=sub<G|y,(x*y)^6*x*y^-1*x*y*(x*y^-1)^2*x*y*x*y*x>;
// L2(16):4.
M3:=sub<G|y,(x*y*x*y*x*y^-1)^2*(x*y^-1*x*y)^2*x>;
// 2^4:(A5 x 3).2.
M4:=sub<G|x,y^-1*(x*y)^5*(x*y^-1)^4>;
// 2 x L2(17).
M5:=sub<G|x,y*(x*y*(x*y^-1)^3)^2*x*y^-1*x*y*x*y>;
// (3 x M10):2.
M6:=sub<G|x,y*x*y^-1*x*y*x*y*(x*y^-1)^4*x*y>;
// 3^{2+1+2}:8.2.
M7:=sub<G|x,(x*y)^5*x*y^-1*x*y*x*y^-1*(x*y)^4>;
// 2^{1+4}.S5.
M8:=sub<G|x,y*x*y^-1*(x*y)^6*(x*y^-1)^2*x*y>;
// 2^{2+4}.(S3 x S3).
M9:=sub<G|x,(x*y*x*y^-1)^2*(x*y)^6*(x*y^-1)^2>;
// 19:18.

H1:=sub<G|y,(x*y)^4*(x*y^-1)^3*x>;
// L2(16).
H2:=sub<G|y,(x*y)^7*(x*y^-1)^2*x*y*x*y*x>;
// L2(16):4.
H3:=sub<G|(x,y)^3,(x*y)^3*x*y^-1*x*y*x*y>;
H3a:=sub<G|(x,y),(y*x)^4*y^-1*x*y*x*y^-1*(x*y)^6*(x*y^-1)^2>;
// L2(19).

/*
Proofs of correctness.
First one: coset enumeration over K = C24.
The following should work.
24*Index(G,K:Print:=2,Hard:=true,CosetLimit:=6*10^6,Grain:=4*10^5);
24*Index(G,K:Print:=2,Hard:=true,CosetLimit:=7*10^6,Grain:=4*10^5); // Better.

Second one:
Coset enumeration over H, whose image in J3:2 is C_G(x), of index 20520.
Coset enumeration gives the expected index.
Relations x^2=y^3=(x*y*(x*y*x*y^-1)^2)^4=1 show that G/G' = 2 and that
G = <x,x^y,x^(y^2)> [so G is generated by conjugates of x].
Generators x and y*(x,y)^4 of H visibly centralise x in the above presentation.
The relations x^2=y^3=(x*y*x*y*x*y^-1*(x*y*x*y*x*y^-1*x*y^-1)^2)^2=1 show
that the third generator of H also centralises x [proof exercise].
Since J3:2 has trivial Schur multiplier this gives the result.
[NB: the central 3 of 3"J3 is inverted by the outer 2.]
The following should work.
Index(G,H:Print:=2,Hard:=true,CosetLimit:=3*10^6,Grain:=4*10^5);

These should work too.
Index(G,M1:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=10);
Index(G,M2:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=4*10^6);
Index(G,M3:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=2*10^6);
Index(G,M4:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=15*10^5);
Index(G,M5:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=3*10^6);
Index(G,M6:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=2*10^6);
Index(G,M7:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=6*10^6);
Index(G,M8:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=3*10^6);
Index(G,M9:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=2*10^6);

Index(G,H1:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=10^6);
Index(G,H2:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=3*10^6);
Index(G,H3:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=10^6);
Index(G,H3a:Print:=2,Hard:=true,Grain:=10^6,CosetLimit:=4*10^6);
*/