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

Let k be odd, and n an odd multiple of 3. Although this can also be deduced from known results, we provide a new proof that Ck ⋊ C8 and (Cn × C3) ⋊ C8 do not have the Directed Cayley Isomorphism (DCI) property. When k is prime, Ck ⋊ C8 had previously been proved to have the Cayley Isomorphism (CI) property. To the best of our knowledge, the groups Cp ⋊ C8 (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 provides a new proof of the result (which follows from known results but was not explicitly published) that no group with an element of order 8 has the DCI property. One piece of our proof is a new result that may prove to be of independent interest: we show that if a permutation group has a regular subgroup of index 2 then it must be 2-closed.