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