Group with a given presentation, finite or infinite

Question

Consider the group with following presentation,

$$G=\langle s,t : s^2=1, (st)^{3}=1\rangle$$

Is this group finite or infinite?

I tried to manipulate the relations and could only get $(ts)^3=1$. I don't know how to proceed further. Any hints?

Answer 1

Hint: Instead of taking $s$ and $t$ as generators, take $s$ and $st$ as generators. How else can you describe the group then?

More details are hidden below.

Writing $u=st$, we have $G=\langle s,u\mid s^2=1, u^3=1\rangle$. But this just means that $G$ is the free product of a cyclic group of order $2$ (generated by $s$) and a cyclic group of order $3$ (generated by $u$). In particular, $G$ is infinite, because for instance there are infinitely many distinct reduced words of the form $sususu\dots$.