Characterization of some simple groups by the multiplicity pattern

Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. Let cd(G) be the set of all character degrees of G. For each positive integer d, the multiplicity of d in G is defined to be the number of irreducible characters of G having the same degree d. The multiplicity pattern mp(G) is the vector whose first coordinate is {pipe}G: G′{pipe} and for i ≥ 1, the (i + 1)-coordinate of mp(G) is the multiplicity of the ith-smallest nontrivial character degree of G. In this paper, we show that every nonabelian simple group with at most 7 distinct character degrees is uniquely determined by the multiplicity pattern.