Index of a subgroup with GAP

Question

Let $G=\langle x,y\ |\ x^2=y^4,\ xyxy^6=1 \rangle$ be a finite presentation group and $H=\langle x^4,y^4,xyx^{-1}y^{-1}\rangle$ be a subgroup of $G$. How could we obtain the index of $H$ in $G$ with GAP?

Answer 1

Try this:

gap> f := FreeGroup("x", "y");
<free group on the generators [ x, y ]>
gap> g := f / [f.1^2 * f.2^-4, f.1 * f.2 * f.1 * f.2^6];
<fp group on the generators [ x, y ]>
gap> x := g.1;
x
gap> y := g.2;
y
gap> h := Group(x^4, y^4, x *  y * x^-1 * y^-1);
Group([ x^4, y^4, x*y*x^-1*y^-1 ])
gap> Index(g, h);
2

See the GAP reference in the chapter about finitely presented groups about groups expressed by generators and relations.