Almost Simple Groups With At Most Nine Special Irreducible Character degrees

Let a, b, c, and d be pairwise relatively prime integers greater than 1 and cd(G) be the set of all irreducible complex character degree of a finite group G. In this paper, by using the Classification of Finite Simple Groups we find all almost simple groups G with cd(G)⊆{1,a,b,c,d,ac,ad,bc,bd}. In particular we show that there exists an integer f≥ 2 such that{2f −1,2f}⊆ cd(G).