Let $G(4, 7)$ be an abstract group with the presentation
$$\langle a,b,c | a^2 = b^2 = c^2 = 1, (ab)^4 = (bc)^4 = (ca)^4 = 1, (acbc)^7 = (baca)^7 = (cbab)^7 = 1 \rangle $$
[Richard Schwartz][1] considered a representation of $G(4,7)$ into $PU(2,1)$ and extensively studied the image group denoted $\Gamma(4,7)$, for instance, the topology of the complex hyperbolic surface $H^{2}_{\mathbb{C}}/\Gamma(4, 7)$. But I'm more curious about $G(4,7)$ itself.

> Is $G(4,7)$ a Coxeter group?

> Whether $G(4,7)$ is a torsion group in which every element has finite order?



  [1]: https://link.springer.com/article/10.1007/s00222-002-0245-8