Cyclic group given a group action

Question

I have been solving some old exam questions to prepare for my own exam, but I have been unable to solve the following question. I am unsure of where to start and would therefore like some hints on where to begin.

Let $(G, ·)$ be a group. Given is a group action $s : G → S_4$ having the property that $s(f) = (12)$ and $s(g) = (34)$ for certain elements $f,g ∈ G$. Can $(G,·)$ be a cyclic group?

Answer 1

If $G$ were generated by $x$, then $x^n=g$ and $x^k=f$ for some $n,k\in \mathbb{N}$. Then $s(x^n)=(34)$ and $s(x^k)=(12)$. What's wrong with that?

Answer 2

Assume $G$ is cyclic. Since $s$ is a homomorphism from $G$ to $S_4$, the subgroup of $S_4$ that is the image of $G$ under $s$ must also be cyclic. But $S_4$ has no cyclic subgroup that contains $(1,2)$ and $(3,4)$. Hence it is not possible that $G$ is cyclic.