  In this article we classify the irreducible representations of finite
classical groups $G$ of Lie type of rank $\ell$ with dimension at most a
constant proportional to $\ell^4$ according to whether $G$ is of type
$A_{\ell}$ or not. The dimensions of those representations are explicitly
computed by hand. We conclude the work by addressing certain cases of the
inverse Galois problem via the foregoing classification.
