Let acd(G) and acs(G) denote the average irreducible character degree and the average conjugacy class size, respectively, of a finite group G. The object of this paper is to prove that if acd(G) < 2 (p + 1)/(p + 3), then G = Op(G) × Op′(G), and that if acs(G) < 4p/(p + 3), then G = Op(G) × O pp'(G) with Op(G) abelian, where p is a prime.