15.47 (2002)

Let $M < G \leqslant \text{Sym}(\Omega)$, where $\Omega$ is finite, be such that $M$ is transitive on $\Omega$ and there is a $G$-invariant partition $\mathcal{P}$ of $\Omega \times \Omega \setminus \{(\alpha, \alpha) \mid \alpha \in \Omega\}$ such that $G$ is transitive on the set of parts of $\mathcal{P}$ and $M$ fixes each part of $\mathcal{P}$ setwise. (Here $\mathcal{P}$ can be identified with a decomposition of the complete directed graph with vertex set $\Omega$ into edge-disjoint isomorphic directed graphs.) If $G$ induces a cyclic permutation group on $\mathcal{P}$, then we showed (Trans. Amer. Math. Soc., 355, no. 2 (2003), 637–653) that the numbers $n = |\Omega|$ and $k = |\mathcal{P}|$ are such that the $r$-part $n_r$ of $n$ satisfies $n_r \equiv 1 \pmod k$ for each prime $r$. Are there examples with $G$ inducing a non-cyclic permutation group on $\mathcal{P}$ for any $n, k$ not satisfying this congruence condition?