Almost simple groups with no product of two primes dividing three character degrees

Let Irr(G) denote the set of complex irreducible characters of a finite group G, and let cd(G) be the set of degrees of the members of Irr(G). For positive integers k and l, we say that the finite group G has the property P-k(l) if, for any distinct degrees a(1), a(2), ..., a(k) is an element of cd(G), the total number of (not necessarily different) prime divisors of the greatest common divisor gcd (a(1), a(2), ..., a(k)) is at most l - 1. In this paper, we classify all finite almost simple groups satisfying the property P-3(2). As a consequence of our classification, we show that if G is an almost simple group satisfying P-3(2), then vertical bar cd(G)vertical bar <= 8.