A classification of vertex-reversing maps with Euler characteristic coprime to the edge number

A map is \emph{vertex-reversing} if it admits an arc-transitive automorphism group with dihedral vertex stabilizers. This paper classifies solvable vertex-reversing maps whose edge number and Euler characteristic are coprime. The classification establishes that such maps comprise three families: $\D_{2n}$-maps, $(\ZZ_{m}{:}\D_{4})$-maps, and $(\ZZ_{m}.\S_4)$-maps, where $m$ is odd. Our classification is based on an explicit characterization obtained of finite almost Sylow-cyclic groups, associated with a shorter proof and explicit description of generators and relations.