We are working in the symmetric group of 4, so I called it S4.
For example, how would you go about finding the other permutations of (12)(34) other than itself, identity, (12), (34)?
These are 4 and by conjugacy classes, we observe that there should be 8 permutations that commute with (12)(34). How would you figure them out? S4 is a big set so you cannot try them all out I guess there should be a method.
I have been looking but I cannot find or understand any method so please if you could make it simple for me. Thanks.
I'm not sure of an efficient method to find all of them. However, one way that works in this case is to recall that if $\sigma\in S_4$, then $\sigma(12)(34)\sigma^{-1}=(\sigma(1)\sigma(2))(\sigma(3)\sigma(4)),$ so if $$\sigma(1)=3,\sigma(2)=4,\sigma(3)=1,\sigma(4)=2,$$ then $\sigma$ works. This tells us another element of the centralizer is $(13)(24).$ To find the others we can just look at the left coset $$(13)(24)\cdot\left<(12),(34)\right>.$$