Prove that every group of order 6 is isomorphic to $C_6$ or $S_3$
So I've already been able to prove this (without what's written below) but I've been told that there's a way to prove this as well using Cauchy's Theorem and the following theorem:
Theorem: The action $G \to S(G/H)$ of G on $G/H$ is a homomorphism with kernel $\bigcap_{x\in G}xHx^{-1}$.
I've been trying to understand how exactly to do so but I get lost when trying to understand how exactly the theorem above helps with anything since the question in general doesn't seem to have anything to do with kernels or subgroups. Do I need to look for a subgroup $H$ such that $S(G/H)$ becomes equivalent to $S_3$ or something? Can someone help?