Dihedral groups of order $2pq$ or $2pqr$ are DCI

A group has the (D)CI ((Directed) Cayley Isomorphism) property, or more commonly is a (D)CI group, if any two Cayley (di)graphs on the group are isomorphic via a group automorphism. That is, $G$ is a (D)CI group if whenever $\mathrm{C}\mathrm{a}\mathrm{y}(G,S)\cong \mathrm{C}\mathrm{a}\mathrm{y}(G,T)$, there is some $\delta \in \mathrm{A}\mathrm{u}\mathrm{t}(G)$ such that $S^{\delta }=T$. (For the CI property, we only require this to be true if $S$ and $T$ are closed under inversion.) Suppose $p,q,r$ are distinct odd primes. We show that $D_{2pqr}$ is a DCI group. We present this result in the more general context of dihedral groups of squarefree order; some of our results apply to any such group, and may be useful in future toward showing that all dihedral groups of squarefree order are DCI groups.