Prove that a normal subgroup $G$ of $S_4$ with $(12)\in S_4$ is equivalent to the entire group $S_4$

Question

Consider a normal subgroup $G$ of $S_4$.

The simple transposition $(12)\in G$. Prove that $G=S_4$.

---

I have already proved that the above case implies that $(12),(23),(34)\in G$. These are all the simple transpositions in $S_4$. I also know that every permutation $\sigma\in S_4$ can be written as a product of the simple transpositions.
Can I use this information to show that $G=S_4$?