/*
Fi22 presented on its standard generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^13,
(x*y)^11,
// (x*y^2)^21, // [R1]
(x*y*x*y*x*y^-3)^5, // [R2]
(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^2*x*y^5)^2
>;
a:=x;b:=y;
// NB: We need R1 or R2, but we do not need both.

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: see http://www.mat.bham.ac.uk/atlas/v2.0/spor/F22/mag/2F22G1-P1.M

Index(G,H:Print:=2,Hard:=true,Grain:=300000,CosetLimit:=10^6);
works with R2, R1 or R1+R2.
*/