/*
2"Fi22 presented on its standard generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^13,
(x*y)^11,
(x,y)^3,
(x,y^2)^3,
(x,y^3)^3,
(x,y^4)^2,
(x,y^5)^3,
(x,y*x*y^2)^3,
(x,y^-1*x*y^-2)^2,
(x,y*x*y^5)^2,
// (x,y^-1*x*y^-5)^3, // Redundant.
(x,y^2*x*y^5)^2
>;
A:=x;B:=y;

H:=sub<G|x,x^(y^4),x^(y^-4),y*x*y^-1*x*y,y^2*x*y^-2*x*y^2,y^3*x*y^-3*x*y^3,
y^5*x*y^-5*x*y^5>;
// H = C_G(x), of index 3510.

/*
Proof:
H visibly centralises x.
[Relations x^2=[x,y]^3=[x,y^2]^3=[x,y^3]^3=[x,y^4]^2=[x,y^5]^3=1.]
G is perfect and generated by the conjugates of x. [Rels x^2=y^13=(xy)^11=1.]
Coset enumeration shows that H has the index 3510 that we expect for the
knwon image Fi22.
Thus G is Fi22, 2"Fi22, 3"Fi22 or 6"Fi22 [with the generators of G mapping onto
standard generators of Fi22].
3"Fi22 is not an image of G [the relations x^2=y^13=(xy)^11=1 take care
of this], but 2"Fi22 is an image.
The central involution is:
(x*y*x*y*x*y^-3)^5=(x*y^2)^21=(x*y^5)^15=(x*y*x*y^2)^15.

The following works.
Index(G,H:Print:=2,Hard:=true,Grain:=300000,CosetLimit:=10^6);

We have not carried out successful enumerations over other subgroups,
so none are given.
*/