I have $G = \langle x,y,z \mid x^2y^4z^3, x^4y^2, x^2y\rangle $.
Now, I have managed to show that its abelianization is isomorphic to $\mathbb{Z}\oplus \mathbb{Z}_3$ but I am really stumped as I am trying to find a homomorphism of $G$ onto $S_3$ but I don't really have a clue where to start.
I considered von Dyck's theorem but I am not sure if it can be applied here as the relation set for $G$ is not a subgroup of the relation set for $S_3$.
Any hints are much appreciated.