/*
Presentation for 2F4(2)' on its standard generators.
Length 143, 6 relations.
*/

G<x,y>:=Group<x,y|
x^2,
y^3,
(x*y)^13,
(x,y)^5,
(x,y*x*y)^4,
(x*y*x*y*x*y*x*y*x*y^-1)^6
>;
a:=x;b:=y;

H:=sub<G|x,y*(x,y)^2,(x,y*x*y)^2,(x,y^-1*x*y^-1)^2>;
// Centralises x, has index 3510 [so not whole of invn centraliser].
K:=sub<G|x*y>;
// C13.
T:=sub<G|>;
// Trivial subgroup.

// Maximal subgroups.
// Ordering does NOT necessarily correspond to ordering of Max1-Max8 used
// elsewhere in this ATLAS.
M1:=sub<G|y,(x*y)^3*(x*y^-1)^4*x*y*(x*y^-1)^2*x*y*x*y*x>;
M2:=sub<G|y,x*y*x*y*x*y^-1*x*y^-1*x*y*x*y*x>;
M3:=sub<G|x,y^-1*x*y*x*y*x*y^-1*x*y*x*y*x*y*x*y^-1*x*y^-1>;
M4:=sub<G|y,x*y*x*y*x*y*x*y^-1*x*y*x*y*x*y^-1*x*y*x*y*x*y*x>;
M5:=sub<G|y,x*y*x*y*x*y*x*y*x*y^-1*x*y^-1*x*y*x*y*x*y^-1*x*y*x*y^-1*x>;
M6:=sub<G|x,y*x*y*x*y*x*y^-1*x*y>;
M7:=sub<G|x,y*x*y^-1*x*y*x*y^-1*x*y^-1*x*y>;
M8:=sub<G|x,y^(x*y)>;

// Some other subgroups.
H1:=sub<G|y,x*y*x*y*x*y^-1*x>;
H2:=sub<G|y,x*y*x*y*x*y*x*y*x*y^-1*x*y*x*y*x*y*x*y*x>;
H3:=sub<G|x,y*x*y^-1*x*y*x*y^-1*x*y,(x,y*x*y)^2>;
H4:=sub<G|x,y*x*y^-1*x*y*x*y^-1*x*y,(x,y*x*y)^2,y*x*y*x*y*x*y*x*y*x*y^-1*x*y*x*y*(x,y*x*y*x*y*x*y*x*y*x*y^-1*x*y*x*y)^2>;
H5:=sub<G|y,x*y*x*y*x*y^-1*x*y*x*y*x*y^-1*x*y*x*y*x>;
H6:=sub<G|y,x^(y*x)>;


/*
Coset enumerations to establish the order of G.
Over K, T or H.
[`Usual' checks must be carried out before one can state that enumeration
over H is a valid method of determining the order of G.]

K = C_{13}, T = 1, 
H centralises x and in the known image
|G:H| = 3510 (so that H is only half the involution centraliser).

The following should work.

Index(G,H:Hard:=true,Print:=2,Grain:=10^4,CosetLimit:=10^4);
13*Index(G,K:Hard:=true,Print:=2,Grain:=5*10^5,CosetLimit:=2*10^6);
Index(G,T:Hard:=true,Print:=2,Grain:=5*10^5,CosetLimit:=19*10^6);

Index(G,M1:Hard:=true,Print:=2,Grain:=10^5,CosetLimit:=10^5);
Index(G,M2:Hard:=true,Print:=2,Grain:=10^5,CosetLimit:=10^5);
Index(G,M3:Hard:=true,Print:=2,Grain:=10^5,CosetLimit:=10^5);
Index(G,M4:Hard:=true,Print:=2,Grain:=10^5,CosetLimit:=10^5);
Index(G,M5:Hard:=true,Print:=2,Grain:=10^5,CosetLimit:=10^5);
Index(G,M6:Hard:=true,Print:=2,Grain:=10^5,CosetLimit:=10^5);
Index(G,M7:Hard:=true,Print:=2,Grain:=10^5,CosetLimit:=10^5);
Index(G,M8:Hard:=true,Print:=2,Grain:=10^5,CosetLimit:=10^5);
*/