Groups Having a Character of Maximal Degree

Abstract

Let $G$ be a group, let $d$ be a character degree, and let $e$ be the integer so that $|G|=d(d+e)$. It has been shown when $e>1$ that $|G|\le e^4-e^3$. In this paper, we consider the groups where $|G|=e^4-e^3$. It is known that $e$ must be a power of a prime. We classify the groups where $e$ is a prime and where $e$ is $4$, $9$, and $25$. In so doing, we find a new nonsolvable Camina pair.