Embedding a Praeger-Xu graph into a surface

Rotary maps (orientably regular maps) are highly symmetric graph embeddings on orientable surfaces. This paper classifies all rotary maps whose underlying graphs are Praeger-Xu graphs, denoted $\mathrm{C}(p,r,s)$, for any odd prime $p$ that does not divide $r$. Our main result establishes a one-to-one correspondence between the isomorphism classes of these maps and the multiplicity-free representations of the dihedral group ${\mathrm{D}}_{2r}$ over the finite field ${\mathbb{F}}_p$. This work extends a recent classification for the case where $p=2$.