  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.
