Prove that there is no simple group of order $216=2^{3}\cdot 3^{3}$.
I feel like I am supposed to use the Index Theorem here but when I use Sylow's Third Theorem I have that $n_{3}\in\{1,4\}$. I am not sure how to rule out the possibility that $n_{3}=4$. If I could then could use the fact that the unique Sylow 3-subgroup is normal in G to factor it and apply the Index Theorem. Any suggestions?