The Cayley isomorphism property for the group $C^5_2\times C_p$

A finite group $G$ is called a DCI-group if two Cayley digraphs over $G$ are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group $C_2^5\times C_p$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$. Together with the previously obtained results, this implies that a group $G$ of order $32p$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$ and $G\cong C_2^5\times C_p$.