I was working towards proving $A_5$ is the only nontrivial normal subgroup of $S_5$. To do this, I wanted to find a set of representatives of conjugacy classes of $S_5$, and their respective orders.
With some research, I found that $1, (12), (123), (12)(34), (1234), (12)(345), (12345)$ are a set of representatives, with orders $1,10,20,15,30,20,24$. What is the most efficient way to determine something like this by hand, without taking an element $\sigma$, conjugating it by all $120$ elements of $S_5$, and then doing it again with a new element which wasn't in the orbit of $\sigma$ until the set is exhausted?