I'm trying to show that if $p\ge3$ is prime, then
There is no extension $K$ of the field of $p$-adic numbers $\mathbb Q_p$ with Galois group $S_4$.
I know that $K$ must have a subextension $K'$ that is unramified over $\mathbb Q_p$, and that $\operatorname{Gal}(K' /\mathbb Q_p)$ must be cyclic. From this I can deduce that the inertia subgroup $I$ is either $S_4$ or $A_4$, since it must be a normal subgroup of $S_4$ with cyclic quotient.
At this point I'm stuck. I've tried playing around with higher ramification groups, using the fact that $G_1$ is the unique Sylow $p$-subgroup of $I$, and I think I can use this to show the $p=3$ case, but haven't been able to get further than this.
I would very much appreciate a hint!