Let $ G $ be a finite group of order $ 2^4\times 3\times 7\times 13$. If $13 $-Sylow subgroup of $ G $ is not normal then $ G $ has 14 Sylow $13$-subgroups. Then $ G$ is $2 $-transitive on the set of Sylow subgroups. Is it possible for $ G $ or always $ G $ has a normal Sylow subgroup?
Any comments or hints are highly appreciated.