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

Abstract: Let $(G,{\rho_0, \rho_1, \rho_2})$ be a string C-group of order $4p^m$ with type ${k_1, k_2}$ for $m \geq 2$, $k_1, k_2\geq 3$ and $p$ be an odd prime. Let $P$ be a Sylow $p$-subgroup of $G$. We prove that $G \cong P \rtimes (\mathbb{Z}_2 \times \mathbb{Z}_2)$, $d(P)=2$, and up to duality, $p \mid k_1, 2p \mid k_2$. Moreover, we show that if $P$ is abelian, then $(G,{\rho_0, \rho_1, \rho_2})$ 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 $m \geq 3$.