A note on arc-transitive circulant digraphs

We prove that, for a positive integer n and subgroup H of automorphisms of a cyclic group Z of order n, there is up to isomorphism a unique connected circulant digraph based on Z admitting an arc-transitive action of Z ⋊ H. We refine the Kovács–Li classification of arc-transitive circulants to determine those digraphs with automorphism group larger than Z ⋊ H. As an application we construct, for each prime power q, a digraph with q – 1 vertices and automorphism group equal to the semilinear group ΓL(1, q), thus proving that ΓL(1, q) is 2-closed in the sense of Wielandt.