Chiral polytopes of order $2p^m$

Abstract

Let $\mathcal{P}$ be a chiral polytope with type $\{k_1, k_2\}$ and $G=Aut(\mathcal{P})$. Suppose $|G|=2p^m$, where $k_1, k_2\geq 3$ and $p$ is an odd prime. Let $P$ be a Sylow $p$-subgroup of $G$. We prove that $G \cong P \rtimes \mathbb{Z}_2$, $d(P)=2$, $P' \ne 1$(so $m \geq 3$) and up to duality, $\{k_1, k_2\}=\{p^{l_1}, 2p^{l_2}\}$ for some integral $l_1, l_2 \geq 1$. Moreover, we show that $\mathcal{P}$ is tight $(k_1k_2=2p^m$) if and only if $P$ is metacyclic group. Furthermore, if $m=3$ or $4$, then $\mathcal{P}$ must be tight, and if $m \geq 5$, where either $m$ is odd, or $m$ is even and $m \geq p+3$, there exists a non-tight chiral polytope $\mathcal{P}$.