Showing an Isomorphism between question group of $S_4$ and $D_6$

Question

I have a subgroup $N$ of $S_4$, where $ N = [1, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)] $ I need to explain whether quotient group $G/N$ is isomoprhic to either $C_6$ or $D_6$ (no proof required, just an explanation to why its isomorphic to one and not the other). Now i know its $D_6$ as N doesn't have a generator element and is not cyclic but this is a weak explanation, I can't spot any other differences.

Answer 1

Hint $S_4$ doesn't have any element of order $6$.

Answer 2

Hint: Suppose there was a surjective homomorphism $\varphi\colon S_4\to C_6$ with $H$ as the kernel. What would the preimage of a generator of $C_6$ have to satisfy?

Answer 3

Suppose $G/N\simeq C_6$. Then $G/N$ contains a unique, and therefore normal, subgroup of order 2. Let $H \le G$ be the pre-image of this group under the natural homomorphism $G \to G/N$. Then $H$ is a normal Sylow-2-subgroup of $G$ which contradicts the structure of $S_4$.