/*
J3 presented in terms of its standard generators.
*/

G<x,y>:=Group<x,y|
x^2,
y^3,
(x,y)^9,
(x*y)^19,
((x*y)^6*(x*y^-1)^5)^2,
((x*y*x*y*x*y^-1)^2*x*y*x*y^-1*x*y^-1*x*y*x*y^-1)^2,
x*y*x*y*(x*y*x*y^-1)^3*x*y*x*y*(x*y*x*y^-1)^4*x*y^-1*(x*y*x*y^-1)^3,
((x*y)^3*(x*y*x*y^-1)^2)^4
>;

a:=x;b:=y;

K1:=sub<G|x*y>;
// C19, of order 19 and index 2643840.
K2:=sub<G|x,y^(x*y)>;
// L2(16), of order 4080 and index 12312.

H:=sub<G|x,y*(x,y)^4,y*x*y*((x*y^-1)^3*x*y*x*y)^3,(y*x*y*x*y^-1*x*y^-1*x)^2*y*(x*y^-1*x*y^-1*x*y*x*y)^2,y*x*y*x*y^-1*x*y*(x,y*x*y*x*y^-1*x*y)^2>;
// H = C_G(x) = 2^{1+4}:A5, of order 1920 and index 26163.


// Maximal subgroups [L2(19)s in same order as those that are produced from
// the word programs J3G1-max2/3W1].
M1:=sub<G|x,(y*x)^5*y^-1*x*y*x*y>;
// L2(16):2, of order 8160 and index 6156.
M2:=sub<G|x,y^-1*x*y*(x*y^-1*x*y*x*y)^2>;
// L2(19), of order 3420 and index 14688.
M3:=sub<G|x,(y*x*y*x*y^-1*x)^2*y*x*y^-1>;
// L2(19), of order 3420 and index 14688.
M4:=sub<G|y,x*y*x*y*x*y^-1*x*(y^-1*x*y*x)^3*y*x*y*x*y^-1*x>;
// 2^4:(3 x A5), of order 2880 and index 17442.
M5:=sub<G|x,y^-1*x*y*x*y*x*y^-1*x*y*x*y*x*y^-1>;
// L2(17), of order 2448 and index 20520.
M6:=sub<G|x,y^(x*y*x*y^-1)>;
// (3 x A6):2b, of order 2160 and index 23256.
M7:=sub<G|x,y*x*y*(x*y^-1)^2*(x*y)^5*(x*y^-1)^2*x*y>;
// 3^2.(3 x 3^2):8, of order 1944 and index 25840.
M8:=sub<G|x,y*x*y*x*y^-1*x*y*(x*y^-1)^3*x*y*x*y*x*y^-1*x*y>;
// 2^{1+4}:A5, of order 1920 and index 26163.
M9:=sub<G|x,(x*y)^3*(x*y^-1)^2*x*y*(x*y^-1)^3*(x*y)^3>;
// 2^{2+4}:(3 x S3), of order 1152 and index 43605.


/*
Enumeration required to establish the order of G is done on the cosets
of K1 = <x*y>, which is cyclic of order 19.
The following should work:
19*Index(G,K1:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=4*10^6);

Other enumerations should work as follows:
Index(G,H:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=10^6);
Index(G,K2:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=10^6);
Index(G,M1:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=10^6);
Index(G,M2:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=10^6);
Index(G,M3:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=10^6);
Index(G,M4:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=10^6);
Index(G,M5:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=10^6);
Index(G,M6:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=10^6);
Index(G,M7:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=4*10^6);
Index(G,M8:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=2*10^6);
Index(G,M9:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=4*10^6);
*/