Quasirandom and quasisimple groups

Fix $\epsilon >0$. We say that a finite group $G$ is $\epsilon$-quasirandom if every nontrivial irreducible complex representation of $G$ has degree at least $\mathrm{\mid }G{\mathrm{\mid }}^{\epsilon }$. In this paper, we give a structure theorem for large $\epsilon$-quasirandom groups, and we completely classify the $\frac{1}{5}$-quasirandom groups.