/*
O10+(2):2 presented on its standard generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^16,
// (x*y)^45, // R1: Redundant.
(x,y)^3, // R2: Redundant but useful.
(x,y^2)^2,
(x,y^3)^3,
(x,y^4)^2,
(x,y^5)^2,
(x,y^6)^2,
(x,y^7)^2,
(x*y^8)^4,
(x*y*x*y^-4)^10, // R3: Redundant but useful.
(x*y^2*x*y^2*x*y^-1)^9
>;


H1:=sub<G|x,y*(x,y),x^(y^2),x^(y^-2),y^3*(x,y^3),x^(y^4),x^(y^-4),x^(y^5)>;
H2:=sub<G|x,x^(y^2),x^(y^-2),y^3*(x,y^3),x^(y^4),x^(y^-4),x^(y^5),x^(y^-5)>;
// H1 = H2 = C_G(x), of index 496.

M2:=sub<G|[x^(y^i):i in [-4,-3,-2,-1,0,1,2,3,4]]>;
// 2 x S8(2), of index 496.
M3:=sub<G|[x^(y^i):i in [-5,-4,-2,-1,0,1,2,3,4]]>;
// 2^8:O8+(2), of index 527.
K1:=sub<G|(x,y^2),x*y*x*y^2*x*y^-4>;
// 2^10:L5(2), of index 4590.


/*
Demonstration of correctness (without R1,R2,R3):
Relations x^2=y^16=(xy^2xy^2xy^-1)^9=1 demonstrate that |G:G'| = 2,
and also force G to be generated by conjugates of x:
indeed, G = <x^(y^i):0\le i \le 15>.
(The extra relation [x,y^5]^2=1 also forces G' to be perfect.)

Relations x^2=[x,y^2]^2=[x,y^3]^3=[x,y^4]^2=[x,y^5]^2=1 show that
H2 centralises x.

Coset enumeration over H2 gives the index 496, what we expect in the known
image O10+(2):2.

[If R2 is added then H1 centralises x, and coset enumeration gives
|G:H1| = 496.]


Checkers for derived series:
============================
AQInvariants(G);
DG:=sub<G|x*y,y*x>;Index(G,DG);
AQInvariants(DG);

Enumeration without R1, R2 or R3:
=================================
Index(G,H1:Hard:=true,Print:=2,CosetLimit:=50000);
Index(G,H2:Hard:=true,Print:=2,Grain:=10^6,CosetLimit:=10^7);

Enumerations with R2 and R3, but excluding R1:
==============================================
Index(G,H1:Hard:=true,Print:=2,CosetLimit:=10000);
Index(G,H2:Hard:=true,Print:=2,CosetLimit:=10000);
Index(G,M2:Hard:=true,Print:=2,Grain:=20000,CosetLimit:=100000);
Index(G,M3:Hard:=true,Print:=2,Grain:=50000,CosetLimit:=300000);

Index(G,K1:Hard:=true,Print:=2,Grain:=10^6,CosetLimit:=60*10^6);
// The last enumeration has not worked yet.

(Enumerations done in Magma2.8.)
*/