Prove that the group $G = \langle x, y \, \mid \, x^2=y^3=(xy)^3=1 \rangle$ is finite.

Question

Let $G = \langle x, y \, \mid \, x^2=y^3=(xy)^3=1 \rangle$. I am trying to prove that $G$ is finite and list all the elements.

Using the fact that $(xy)^2=(xy)^{-1}=y^2x$, I have managed to show that every element of $G$ is of the form $x^iy^jx^k$ or $y^ix^jy^k$. This proves that $G$ is finite. The problem is, some of these elements coincide (for example, $yxy=xy^2x$), so I am not sure what the order of $G$ is.

Is there a "nicer" proof that actually determines $|G|$? In another question, I am asked to find the centralizer of a certain subgroup, so I hope to understand the elements of $G$ better in this part.

Answer 1

Since $(xy)^3$ is equivalent to $x^{-1}(xy)^3x=(yx)^3$, we can just use a Tietze transformation to convert that presentation of $G$ to the one handled here (after relabelling $x\leftrightarrow y$):

Why is $\langle x,y \mid x^3=y^2=(xy)^3=e\rangle$ a presentation of $\mathfrak{A}_4$?