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.