String C-groups of order 4pᵐ

Let (G,\ρ₀, ρ₁, ρ₂\) be a string C-group of order 4pᵐ with type ₁, k₂\ for m ≥ 2, k₁, k₂≥ 3 and p be an odd prime. Let P be a Sylow p-subgroup of G. We prove that G P (Z₂ × Z₂), $d(P)=2$, and up to duality, p k₁, 2p k₂. Moreover, we show that if P is abelian, then (G,\ρ₀, ρ₁, ρ₂\) is tight and hence known. In the case where P is nonabelian, we construct an infinite family of string C-group with type , 2p\ of order 4pᵐ where m ≥ 3.