  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 $m \geq 3$.
