Is there a way to get all the permutations of $S_4$ I need to calculate the determinant of a $4 \times 4$ matrix by "direct computation", so I thought that means using the formula
$$\sum_{\sigma \in S_4} (-1)^{\sigma}a_{1\sigma(1)}\ldots a_{n\sigma(n)}$$
So first I wanted to write down all the permutations of $S_4$ but I've only got 23 out of the 24 and I can't think of the last one. I was wondering if there is a "method" I can use to get all of them (apart from Googling them) and make sure that they are all unique and I've not done the same one twice?
Right now, I have
$$\begin{matrix} () & (34) & (143) & (1243) \\ (12) & (123) & (234) & (3241)\\  (13) & (132) & (243) & (1324) \\ (14) & (124) & (324) & (4231) \\ (23) & (142) & (1234) & (4321) \\ (24) & (134) & (2134) \\ \end{matrix}$$
What one am I missing?
 A: I don't think the cycle structure is particularly helpful for enumerating all permutations. There it is easier to think of a permutation as simply the numbers form 1 to 4 arranged in some order.
All such sequences can be generated systematically by taking first those that have 1, then those that have 2 in the first place, and so on. Within each group, do a similar split on the second place, and proceed recursively. You get:
1,2,3,4
1,2,4,3
1,3,2,4
1,3,4,2
1,4,2,3
1,4,3,2
2,1,3,4
...
2,4,3,1
3,1,2,4
...
4,3,1,2
4,3,2,1

If you use this enumeration to compute determinants, you may notice that it you add the terms in the order (terms from permutations with 1 in the first place) + (terms from permutations with 2 in the first place) + ... + (terms from permutations with 4 in the first place), what you're doing is exactly expansion by minors!
A: First of all, you have repeating permutations. For example: $(243)=(324)$ (the same with the four-cycles... there should only be 6 of them).
You are missing the permutations of the structure $(--)(--)$. For example $(12)(34)$,$(13)(24)$.
