Finding the square root of a permutation cycle

Question

Find all $\sigma \in S_4$ satisfying $\sigma^2 = (1 2)(3 4)$

By just brute forcing it, I can easily find $\sigma = (1 3 2 4)$ or $\sigma = (1423)$. But is there a more mathematical way of finding these permutation cycles?

Answer 1

It is not difficult to determine the orders of each of the elements of $S_4$. The square root of $(12)(34)$ must have order $4$ so you only need to check them.
By the way you should probably change "square" to "square root" in the title.