String C-groups of order $4p^m$

Abstract

Let $(G,\{\ddot{I}{}_0,\ddot{I}{}_1,\ddot{I}{}_2\})$ be a string C-group of order $4p^m$ with type $\{k_1,k_2\}$ for $m\ge 2$, $k_1,k_2\ge 3$ and $p$ be an odd prime. Let $P$ be a Sylow $p$-subgroup of $G$. We prove that $G\cong P⋊({\mathbb{Z}}_2\times {\mathbb{Z}}_2)$, $d(P)=2$, and up to duality, $p\mid k_1,2p\mid k_2$. Moreover, we show that if $P$ is abelian, then $(G,\{\ddot{I}{}_0,\ddot{I}{}_1,\ddot{I}{}_2\})$ is tight and hence known. In the case where $P$ is nonabelian, we construct an infinite family of string C-group with type $\{p,2p\}$ of order $4p^m$ where $m\ge 3$.

arXiv admin note: text overlap with arXiv:2107.02925