Groups with elements of order 8 do not have the DCI property

Abstract

Let $k$ be odd, and $n$ an odd multiple of $3$. We prove that $C_k \rtimes C_8$ and $(C_n \times C_3)\rtimes C_8$ do not have the Directed Cayley Isomorphism (DCI) property. When $k$ is also prime, $C_k \rtimes C_8$ had previously been proved to have the Cayley Isomorphism (CI) property. To the best of our knowledge, the groups $C_p \rtimes C_8$ (where $p$ is an odd prime) are only the second known infinite family of groups that have the CI property but do not have the DCI property. This also shows that no group with an element of order $8$ has the DCI property.