> G<a,b> := Group<a,b|a^2,b^3,(a*b)^7,((a,b)^2*a*b)^6 >;
> SQ := SimpleQuotients(G,1000000: Limit:=10 );
> ChiefFactors(Image(SQ[1][1]));
    G
    |  A(1, 13)                   = L(2, 13)
    1
> P := pQuotient(Kernel(SQ[1][1]), 2, 3 : Print:=1);

Lower exponent-2 central series for $

Group: $ to lower exponent-2 central class 1 has order 2^14

Group: $ to lower exponent-2 central class 2 has order 2^42

Group: $ to lower exponent-2 central class 3 has order 2^70

> ChiefFactors(Image(SQ[2][1]));
G
|  J1
1
> h := SQ[3][1];
> I := Image(h);
> L := LowIndexSubgroups(I,1000);
> [Index(I,l): l in L];
[ 840, 560, 280, 525, 315, 100, 1 ]
> p := CosetAction(I,L[4]);
> S := sub< G | h*p >;
> Index(G,S);
525
> AbelianQuotientInvariants(S);
[ 6, 0 ]