Finite groups with an irreducible character of large degree

Let $G$ be a finite group and $d$ the degree of a complex irreducible character of $G$, then write $\mathrm{\mid }G\mathrm{\mid }=d(d+e)$ where $e$ is a nonnegative integer. We prove that $\mathrm{\mid }G\mathrm{\mid }\le e^4-e^3$ whenever $e>1$. This bound is best possible and improves on several earlier related results.