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

G<x,y>:=Group<x,y|
x^2,
y^3,
(x,y)^9,
((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*(x*y)^19
>;

A:=x;B:=y;


K0:=sub<G|x*y,(x*y)^3*(x*y*x*y^-1)^2>;
// The whole of 3"J3.
K1:=sub<G|x,x^y>;
// D18, of order 18 and index 8372160.
K2:=sub<G|x,y^(x*y)>;
// L2(16), of order 4080 and index 36936.


// `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*(x*y)^19>;
M1a:=sub<G|x,y*(x*y*x*y^-1)^2*(x*y^-1)^2*x*y*x*y>;
// L2(16):2, of order 8160 and index 18468.
M2:=sub<G|x,y^-1*x*y*(x*y^-1*x*y*x*y)^2*(x*y)^-19>;
// L2(19), of order 3420 and index 44064.
M3:=sub<G|x,(y*x*y*x*y^-1*x)^2*y*x*y^-1*(x*y)^-19>;
// L2(19), of order 3420 and index 44064.
M5:=sub<G|x,y^-1*x*y*x*y*x*y^-1*x*y*x*y*x*y^-1*(x*y)^-19>;
M5a:=sub<G|x,(x*y)^4*(x*y^-1)^3*(x*y)^4>;
// L2(17), of order 2448 and index 61560.
M6:=sub<G|x,y^(x*y*x*y^-1)*(x*y)^19>;
// (3 x A6):2b, of order 2160 and index 69768.
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*(x*y)^19>;
M8a:=sub<G|x,(y*x*y^-1*x)^2*(y^-1*x*y^-1*x*y*x*y*x)^2*x^-1*y^-1>;
// 2^{1+4}:A5, of order 1920 and index 78489.



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*(x*y)^-19>;
// 2^4:(3 x A5), of order 2880 and index 52326.
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 130815.


/*
Define z to be (x*y)^-19.
Then the last relation is: ((x*y)^3*(x*y*x*y^-1)^2)^4 = z = (x*y)^-19.
Thus z commutes with K0 = <x*y,(x*y)^3*(x*y*x*y^-1)^2>.
Coset enumeration over K0 shows that z is central. Then G/<z> clearly
has the presentation of J3 given in J3G1-P1.M.
Checking that G is perfect then forces G to be a quotient of 3"J3.
[One uses a favourite representation to show that the image 3"J3 exists.]

AQInvariants(G);
Index(G,K0:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=1000);


A direct approach - enumerating the cosets of K1 = <x,x^y> = D18 - actually
works, but requires much computer power.
18*Index(G,K1:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=22*10^6);


Other enumerations should work as follows:
Index(G,K2:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=10^6);
Index(G,M1:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=3*10^6);
Index(G,M2:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=5*10^6);
Index(G,M3:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=5*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:=2*10^6);
Index(G,M8:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=16*10^6);

Index(G,M1a:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=5*10^6);
Index(G,M4:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=7*10^6);


Index(G,M5a:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=???);
Index(G,M8a:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=???);
Index(G,M9:Print:=2,Hard:=true,Grain:=10^5,CosetLimit:=???);
*/