String C-groups of order 4pᵐ

Let (G, \₀, ₁, ₂\) be a string C-group of order 4pᵐ with type \k₁, 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, \₀, ₁, 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 4pᵐ where m 3.