Group presentation

Question

Given $G = \langle x,y \mid x^{16}=y^{24}=e, x^2=y^3\rangle$, if $G$ is abelian, find its order.

Note: An estimation gives $\left|G\right| \leq 48$, but is there a quotient of $G$ with order $48$? (If so, by von Dyck's theorem one can then deduce $\left|G\right|=48$).

Answer 1

Consider the additive cyclic group $H = \mathbb{Z}/48\mathbb{Z}$ with $x = 3$ and $y = 2$. Note that $x-y$ generates $H$.

Answer 2

Denote $u=xy^{-1}$. Then $x=u^3, y=u^2$, so $G=\langle u\rangle$ is cyclic.