Well, we know all normal subgroups of $S_n$, including the interesting case $n=4$. Using this one can conclude that the cyclic group of order 3 is not a quotient of $S_n$. Is there are more direct way to see this?
$C_3$ is simple, so if there were a surjective homomorphism from $S_n$ onto $C_3$, its kernel would be a maximal subgroup of index 3. The alternating group is also maximal as it has index 2. Can we combine these two fact together to get a contradiction?