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

TL;DR

This paper classifies string C-groups of order 4p^m, revealing their structure, conditions on parameters, and providing new infinite families when the Sylow p-subgroup is nonabelian.

Contribution

It characterizes the structure of string C-groups of order 4p^m and constructs new infinite families for nonabelian Sylow p-subgroups.

Findings

G is isomorphic to P semi-direct product with Z_2 x Z_2

If P is abelian, the group is tight and known

New infinite families of string C-groups with nonabelian P

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 $m \geq 3$.