gr.group theory – How do I find hyperbolic generating triples for a group using GAP?

Let $$G$$ be a finite group and $$x, y, z in G$$. A hyperbolic generating triple for $$G$$ is a triple $$(x, y, z) in Gtimes Gtimes G$$ such that

• $$frac{1}{o(x)}+frac{1}{o(y)}+frac{1}{o(z)} <1$$,
• $$langle x,y,z rangle =G$$, and
• $$xyz=1$$.

The type of a hyperbolic generating triple $$(x, y, z)$$ is the triple $$(o(x), o(y), o(z))$$.

My question is, how can I use GAP to determine these triples for a group and therefore their type? Take $$PSL(2, 7)$$ as an example.

Thanks