  Let $H$ be a finite quasisimple classical group, i.e. $H$ is perfect and
$S:=H/Z(H)$ is a finite simple classical group. We prove in this paper that,
excluding the cases when the simple group $S$ has a very exceptional Schur
multiplier such as $\PSL_3(4)$ or $\PSU_4(3)$, $H$ is uniquely determined by
the structure of its complex group algebra. The proofs make essential use of
the classification of finite simple groups as well as the results on prime
power character degrees and relatively small character degrees of quasisimple
classical groups.
