Nonsolvable groups with no prime dividing three character degrees

We consider nonsolvable finite groups G with the property that no prime divides at least three distinct character degrees of G. We first show that if , where S is a nonabelian finite simple group, and no prime divides three degrees of G, then S≅PSL2(q) with q⩾4. Moreover, in this case, no prime divides three degrees of G if and only if G≅PSL2(q), G≅PGL2(q), or q is a power of 2 or 3 and G is a semi-direct product of PSL2(q) with a certain cyclic group. More generally, we give a characterization of nonsolvable groups with no prime dividing three degrees. Using this characterization, we conclude that any such group has at most 6 distinct character degrees, extending to the nonsolvable case the analogous earlier result of D. Benjamin for solvable groups.