Prove that the conjugacy classes in have sizes 1,12 ,12, 15 and 20..

so to prove the conjugacy class of size 1, its trivial, its just the identity.

to prove the conjugacy class of size 20, i used a lemma saying... All 3 cycles in are conjugate to each other,

to prove the conjugacy class of 15 i used

there are ways to do the first cycle, of x and there are to do the second cycle in x, and to eliminate similar cycle structures we divide by 2.

Not really sure if this actually proves the 15, and i know there are 24 elements left of order 5... (process of elimination, how do i prove they have to be 12 + 12...)??