Low Index Subgroups in GAP

Question

I want to find an algorithm for constructing a coset table for all subgroups of index 14 of triangle group (2,3,7) by using Low index subgroups.

for this I am making a program like this.

f:=FreeGroup(2);

g:=f/[f.1^2,f.2^3,(f.1*f.2)^7];

LowIndexSubgroupsFpGroup(g,14); where 14 is the index of the subgroup in g. However, it gives me Group() Could any one help me finding such an algorithm for.

Answer 1

The following code finds the 9 conjugacy classes of subgroups of index 14, and displays the coset table of the first one. The rows are $f_1, f_1^{-1}, f_2, f_2^{-1}$. For completed coset tables, the inverses are not very interesting, but they can be very useful for incomplete coset tables. I also include the coset table rows written as permutations, since that seems more useful to me.

gap> f:=FreeGroup(2);;
gap> g:=f/[f.1^2,f.2^3,(f.1*f.2)^7];;
gap> hs:=Filtered(LowIndexSubgroupsFpGroup(g,14),h->Index(g,h)=14);; Size(hs);
9
gap> PrintArray( CosetTable( g, hs[1] ) ) ;;
[ [   1,   2,   4,   3,   7,   8,   5,   6,  12,  13,  14,   9,  10,  11 ],
  [   1,   2,   4,   3,   7,   8,   5,   6,  12,  13,  14,   9,  10,  11 ],
  [   2,   3,   1,   5,   6,   4,   9,  11,  10,   7,  12,   8,  13,  14 ],
  [   3,   1,   2,   6,   4,   5,  10,  12,   7,   9,   8,  11,  13,  14 ] ]
gap> List( CosetTable( g, hs[1] ){[1,3]}, PermList );
[ (3,4)(5,7)(6,8)(9,12)(10,13)(11,14), (1,2,3)(4,5,6)(7,9,10)(8,11,12) ]