The Sylow Theorems · Superchallenge problem

A simple group has order $168$. If it had a subgroup of index $k$, the action on the $k$ cosets would give an embedding $G\hookrightarrow S_k$ (the kernel must be trivial, since $G$ is simple and the action is non-trivial for $k>1$), forcing $|G|$ to divide $k!$. What is the SMALLEST integer $k$ for which $k! \ge 168$?