On Cayley representations of finite graphs over abelian p-groups

We construct a polynomial-time algorithm which given a graph $\Gamma$ finds the full set of non-equivalent Cayley representations of $\Gamma$ over the group $D\cong C_p\times C_{p^k}$, where $p\in\{2,3\}$ and $k\geq 1$. This result implies that the recognition and the isomorphism problems for Cayley graphs over $D$ can be solved in polynomial time.