/*
J4 presented on its G2-`standard' generators.
*/

G<x,y,t>:=Group<x,y,t|
x^2,
y^3,
(x*y)^23,
(x,y)^12,
(x,y*x*y)^5,
(x*y*x*y*x*y^-1)^3*(x*y*x*y^-1*x*y^-1)^3,
(x*y*(x*y*x*y^-1)^3)^4,
t^2,
(t,x),
(t,y*x*y*(x*y^-1)^2*(x*y)^3),
(y*t^(y*x*y^-1*x*y*x*y^-1*x))^3,
((y*x*y*x*y*x*y)^3*t*t^((x*y)^3*y*(x*y)^6*y))^2
>;

M1:=sub<G|x,y,(y*t^((y*x)^4*y^-1*x))^3>;
// M1 = 2^11:M24.
M2:=sub<G|t,x,y*x*y*(x*y^-1)^2*(x*y)^3,t^(x*y*x*y^-1*x*y),t^(x*y*x*y^-1)>;
M2a:=sub<G|x,y*x*y*(x*y^-1)^2*(x*y)^3,t^(x*y*x*y^-1)>;
M2b:=sub<G|y*x*y*(x*y^-1)^2*(x*y)^3,x*t^(x*y*x*y^-1)>;
// M2 = M2a = M2b = C_G(t) \cong 2^{1+12}.3"M22.2.

/*
Proof:
Er, ... a little hard ... too large for this file to contain.
One method basically consists of showing that M2 centralises t, and
then enumerating the 3980549947 cosets of this subgroup.
Requires double coset enumeration.
Details in a forthcoming paper (honest).
*/