Nonsolvable groups with three nonlinear irreducible character codegrees

Abstract

For an irreducible character $\chi$ of a finite group $G$, the codegree of $\chi$ is defined as $|G:\ker (\chi )|/\chi (1)$. In this paper, we determine finite nonsolvable groups with exactly three nonlinear irreducible character codegrees, and they are ${\mathrm{L}}_2(2^f)$ for $f\ge 2$, ${\mathrm{PGL}}_2(q)$ for odd $q\ge 5$ or ${\mathrm{M}}_{10}$.