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

G<x,y>:=Group<x,y|
x^2,
y^18,
(x,y^6)^2,
(x,y^7)^2,
(x*y^9)^4, // Redundant.
(x,y)^3,
(x,y^2)^3,
(x,y^3)^3,
(x,y^4)^3,
(x,y^5)^3,
(x,y^8)^3,
(x,y*x*y*x*y^-2*x*y),
(x,y^2*x*y^2*x*y^-4*x*y^2),
((x*y^3)^4*x*y^-4)^2,
(x*y^4*x*y^5*x*y^5)^5,
(x*y*x*y^-3)^8
>;

DG:=sub<G|x,y^2,y*x*y>;

H1:=sub<G|x,(x,y^6),(x,y^-6),(x,y^7),(x,y^-7),(x*y^9)^2,y*(x,y),y^2*(x,y^2),
y^3*(x,y^3),y^4*(x,y^4),y^5*(x,y^5),y^8*(x,y^8)>;
H2:=sub<G|x,(x,y^6),(x,y^-6),(x,y^7),(x,y^-7),y*(x,y),y^2*(x,y^2),
y^3*(x,y^3),y^4*(x,y^4),y^5*(x,y^5),y^8*(x,y^8)>;
// H1 = H2 = C_G(x), of index 3510.

/*
Proof: see http://www.mat.bham.ac.uk/atlas/v2.0/spor/F22/mag/3F22d2G1-P1.M

Enumeration that works.

With (x*y^9)^4.
Index(G,H1:Print:=2,Hard:=true,CosetLimit:=10^6,Grain:=10^5);
*/