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Just want to add that $G$ has a quotient isomorphic to $H:=\mathbb Z_p \ltimes_\varphi \mathbb Z^{p-1}$ where $\varphi(1)\cdot v=Av$ for some $A \in SL_{p-1}(\mathbb Z)$ of order $p$; in particular, $G$ is infinite.
Indeed, such an $A$ exists: take $A$ to be a companion matrix to the polynomial $\frac{x^p-1}{x-1}$. Then any element in $H$ not contained in $\mathbb Z_p {0} \times {0}$ \mathbb Z ^{p-1}$has order$p$, so mapping$x$to$(1,0)$and$y$to$(1,e_1)$will give a well-defined homomorphism which can be shown to be surjective. 1 Just want to add that$G$has a quotient isomorphic to$H:=\mathbb Z_p \ltimes_\varphi \mathbb Z^{p-1}$where$\varphi(1)\cdot v=Av$for some$A \in SL_{p-1}(\mathbb Z)$of order$p$; in particular,$G$is infinite. Indeed, such an$A$exists: take$A$to be a companion matrix to the polynomial$\frac{x^p-1}{x-1}$. Then any element in$H$not contained in$\mathbb Z_p \times {0}$has order$p$, so mapping$x$to$(1,0)$and$y$to$(1,e_1)\$ will give a well-defined homomorphism which can be shown to be surjective.