/*
Sz(8) presented on its standard generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^4,
(x*y)^5,
(x*y^2)^7,
(x*y*x*y^-1*x*y^2)^7, // R1.
// (x,y)^13             // R2.
// (x*y*x*y^2*x*y^-1*x*y^2*x*y^2)^5, // R3.
x*y*x*y^2*x*y^2*x*y*x*y^2*(x,y)^3*(x*y^2*x*y^-1*x*y^2)^2 // R4.
>;

// Maximal subgroups.
M1:=sub<G|y,x*y*x*y^-1*x*y^2*x*y*x*y^2*x>;
M2:=sub<G|x,x*y*x*y^2*x*y*x*y^-1*x*y*x*y>;
M3:=sub<G|y,x^(y^2*x*y*x*y^2*x)>;
M4:=sub<G|x,y^2>;

// Some other subgroups.
H1:=sub<G|x,y*x*y^-1*x*y^2*x*y*x*y^2>; // 2^3:7.
H2:=sub<G|x,x^y>; // D26.
H3:=sub<G|x,x^(y*x*y^2)>; // D10.


/*
Groups presented by x^2=y^4=(xy)^5=(xyy)^7=1 + subsets of {R1,R2,R3,R4}.

R1 R2 R3 R4   Group presented.
 n  n  n  n   Infinite - Known image (2^2"Sz(8))^2 x 41^7:L2(41).
 n  n  n  Y   2"Sz(8) (not on standard generators). - Type P.
 n  n  Y  n   2"Sz(8) (not on standard generators). - Type Q.
 n  n  Y  Y   Sz(8).
 n  Y  n  n   ??? Known image 2.(2"Sz(8) x 2"Sz(8)).
 n  Y  n  Y   Sz(8).
 n  Y  Y  n   2"Sz(8) (not on standard generators). - Type Q.
 n  Y  Y  Y   Sz(8).
 Y  n  n  n   2"Sz(8) (on standard generators).
 Y  n  n  Y   Sz(8). - Best for coset enumeration.
 Y  n  Y  n   Sz(8).
 Y  n  Y  Y   Sz(8).
 Y  Y  n  n   Sz(8).
 Y  Y  n  Y   Sz(8).
 Y  Y  Y  n   Sz(8).
 Y  Y  Y  Y   Sz(8).

Proofs:
If R1 or R4 is present.
Coset enumeration over trivial subgroup or <x,y^2> = D14. (Seems easy.)

Rels R3 (alone) or R2+R3 (but not R1 or R4).
Coset enumeration over <x,y^2> = D14. Use
H:=sub<G|x,y^2>;
14*Index(G,H:Hard:=true,CosetLimit:=10^6,Grain:=100000);

None of rels R1,R2,R3,R4.
Holt & Plesken (J.LMS(2) (1992) 469-480) show that Z^28.Sz(8) is an image,
so this group is infinite.

Rel R2 (alone).
We don't know.
*/