I am looking at a specific group $G$ and trying to find an index 2 subgroup with generators of length 3, such that the generators form a spherical system. Here, an $r$-tuple $[g_1,\ldots,g_r]$ of elements of $G$ is called a spherical system of generators if $g_1,\ldots,g_r$ generate $G$ and we additionally have $g_1\cdots g_r=1$.
So far, I have written code for the group with a maximal two quotient of class 3. I'd be interested to change the class at a later point and see the effect if any.
$\left\langle\ x_{0},...,x_{20}\mid x_{i}x_{i+7}x_{i+14}, x_{i}x_{i+14}x_{i+7}, x_{i}x_{i+3}x_{i+15}\:(i\in\mathbb{Z}_{21})\right\rangle$
f:=FreeGroup(List([1..21],i->Concatenation("x_",String(i mod 21))));;
gens:=GeneratorsOfGroup(f);;
m:=function(a,b) return ((a-1) mod b)+1; end;;
LIST1 :=List([1..21],i->Product(gens{List([i,i+7,i+14],j->m(j,21))}));;
LIST2 :=List([1..21],i->Product(gens{List([i,i+14,i+7],j->m(j,21))}));;
LIST3 :=List([1..21],i->Product(gens{List([i,i+3,i+15],j->m(j,21))}));;
r := Concatenation([ LIST1, LIST2, LIST3 ]);;
g:=f/r;;
epi:=EpimorphismPGroup(g,2,3);;
G:=Image(epi);
Size(G);
p:=PresentationFpGroup(g);
SimplifyPresentation(p);
TzPrintPresentation(p);
Now to find a subgroup H with index 2, I have used
H:=LowIndexSubgroupsFpGroup( g,2);
And so I now need to find the generators and relators of these. So I have,
GeneratorsOfGroup(H[ i ]);
h:=SimplifiedFpGroup(Image(IsomorphismFpGroup(H[i])));
RelatorsOfGroup(h);
I am struggling with the need to go through all H[i] (I think there are 25 in this case). What is also not apparent is whether they form a spherical system of generators. In particular, I cannot see any subgroups that have generators of length 3.
Has anybody got any tips on how to possibly look for this or smarten up the code?
Thanks