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: ${\text{D}}_{2n}$-maps, $({\text{ZZ}}_m:{\text{D}}_4)$-maps, and $({\text{ZZ}}_m\mathrm{.}{\mathrm{\S }}_4)$-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.