Maps, simple groups, and arc-transitive graphs

We determine all factorisations X=AB, where X is a finite almost simple group and A, B are core-free subgroups such that A B is cyclic or dihedral. As a main application, we classify the graphs admitting an almost simple arc-transitive group X of automorphisms, such that has a 2-cell embedding as a map on a closed surface admitting a core-free arc-transitive subgroup G of X. We prove that apart from the case where X and G have socles Aₙ and A₍-₁ respectively, the only such graphs are the complete graphs Kₙ with n a prime power, the Johnson graphs J (n, 2) with n-1 a prime power, and 14 further graphs. In the exceptional case, we construct infinitely many graph embeddings.