Finite groups with special Sylow 2-subgroups

Let $T$ be a Sylow 2-subgroup of a simple group $PSU(3,2^n)$, and $Z$ a proper subgroup belonging to the center of $T$. We shall prove that a simple finite group whose Sylow 2-subgroup is isomorphic to $T/Z$ coincides with $PSU(3,2^n)$. As a consequence we list simple groups that can be represented in the form of a product of two Schmidt groups, i.e., of minimal nonnilpotent groups.