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

TL;DR

This paper characterizes when the group $C_2^5 imes C_p$ is a DCI-group, showing it holds if and only if $p$ is an odd prime, thus advancing understanding of Cayley graph isomorphisms for specific finite groups.

Contribution

It proves that $C_2^5 imes C_p$ is a DCI-group precisely when $p$ is an odd prime, completing the classification for groups of order $32p$.

Findings

$C_2^5 imes C_p$ is a DCI-group if and only if $p eq 2$

Groups of order $32p$ are DCI-groups only when $p eq 2$ and $G ot ightarrow C_2^5 imes C_p$

The result extends the classification of DCI-groups for certain group orders.

Abstract

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 $G\cong C_2^5\times C_p$.