$G$ a group of order 36, $P$ is the only Sylow 3-subgroup, which is normal, prove the existence of a homomorphism $\phi:G\rightarrow S_4$ Specifically the question I was asked was to prove that there was either a normal subgroup of $G$ of index 4 or that there was a non-trivial homomorphism $\phi:G\rightarrow S_4$, i.e., such that $\phi(g)\neq 1$ for some $g$. As $n_3=|G:N_G(P)|$ for any Sylow p-subgroup and $n_3=1,4$, if $n_3=4$ I'm done so I set $n_3=1$ and proved the information I have in the title but I can't get any further. 

If $P$ is the unique Sylow subgroup it must be normal (a conjugate of $P$ would be a different Sylow subgroup).
