# Automorphism group of directed complete graph

Given a directed complete graph on $n$ vertices, is there an efficient algorithm for computing its automorphism group? Is there a nontrivial upper bound on the order of its automorphism group? How about the isomorphism of two such graphs?

For an upper bound on the group size, take $n!$ and remove from it the highest power of 2, but probably that is way too large. A better bound would be the answer to this stackoverflow question.
The best known algorithm needs time $n^{O(\log n)}$, which is better than the best known for general digraphs.
As to an algorithm: There is Tarjan's algorithm for computing strongly connected components. That should certainly be a first step in every such algorithm because every automorphism will permute strongly connected components. In fact in a complete graph the components are linearly ordered so that any automorphism will in fact map all components onto themselves. Moreover: If $\Gamma = \Gamma_1 \sqcup \Gamma_2 \sqcup \ldots \sqcup \Gamma_k$ is the decomposition into components in descending order then by completeness all vertices in $\Gamma_i$ will be have an arrow pointing toward all vertices in $\Gamma_j$ if $i<j$. Therefore $Aut(\Gamma) = Aut(\Gamma_1) \times Aut(\Gamma_2) \times \ldots \times Aut(\Gamma_k)$ which reduces the problem to finding the automorphism group of a strongly connected complete graph.