Calculate the commutator subgroup of $S_4$ So I have been tasked with calculating the commutator subgroup of $S_4$. As a warmup, I was able to calculate the commutator subgroup of $S_3$ through brute force calculations as there were only $6^2$ possibilities. I found that ${S_3}^{'}=\{e,(1\,\,2\,\,3),\,(1\,\,3\,\,2)\}$.
For $S_4$, I clearly do not want to attempt all $24^2$ computations, so what kind of strategy could I employ to get this done in a reasonable amount of time?

First note that all commutators will be even permutations.
Then note that $[ (a, c), (a, b)] = (a, b, c)$, if $a, b, c$ are distinct.
So in $S_{4}'$ you find all the $3$-cycles.
