We determine all factorisations X = AB, where X is a fini
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 Γ admitting an almost simple arc transitive group X of automorphisms, such that Γ 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 An−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.