A family of regular polytopes of order $4p^m$ with type $\{p, 2p\}$

Abstract: In this paper, we construct an infinite families of group $G$ of order $4p^m$ which can be an automorphism group of some regular polytope with type ${p, 2p}$, where $m \geq 3$ and $p$ is an odd prime. For $p=3$, our polytopes are the regular toroidal maps ${3, 6}$.