Geodesic
Groups with Relatively Few Non-Linear Irreducible Characters
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 1451-1458

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In (4), Seitz characterized those finite groups which have exactly one non-linear irreducible character (over the complex numbers). In this paper we are concerned with the general question of what can be deduced about a finite group G if the number of its non-linear irreducible characters m(G) is given. In particular, does the assumption that m(G) is in some sense small when compared with the order |G| impose any restrictions on the structure of G? 
Isaacs, I. M.; Passman, D. S. Groups with Relatively Few Non-Linear Irreducible Characters. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 1451-1458. doi: 10.4153/CJM-1968-146-3
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[1] 1. Higman, G., Suzuki 2-groups, Illinois J. Math. 7 (1963), 79–96. Google Scholar

[2] 2. Isaacs, I. M. and Passman, D. S., A characterization of groups in terms of the degrees of their characters. II, Pacific J. Math., 24 (1968), 467–510. Google Scholar

[3] 3. Landau, E., Uber die Klassenzahl der bindren quadratischen Formen von negativer Discriminante, Math. Ann. 56 (1903), 671–676. Google Scholar

[4] 4. Seitz, G., Finite groups having only one irreducible representation of degree greater than one, Proc. Amer. Math. Soc. 19 (1968), 459–461. Google Scholar

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