Chiral polytopes of order 2p^m

Let P be a chiral 3-polytope with type {k₁, k₂} and G=Aut(P). Suppose |G| = 2pm, where k₁, k₂ ≥ 3 and p is an odd prime. Let P be a Sylow p-subgroup of G. We prove that G ≅ P ⋊ ℤ₂, that the size of a minimal generating set of elements of P is 2, that P′ ≠ 1(so m ≥ 3) and that, up to duality, {k₁, k₂} = {pl₁, 2pl₂} for some integers l₁, l₂ ≥ 1. Moreover, we show that P is tight (k₁k₂ = 2pm) if and only if P is metacyclic group. Furthermore, if m = 3 or 4, then P must be tight, and if m ≥ 5, where either m is odd, or m is even and m ≥ p + 3, there exists a non-tight chiral 3-polytope P.