Suppose we have a group $A_n$ for some $n$ (maybe take $A_5$ as an example).
We find the conjugacy classes of $S_n$ which are determined by cycle type. Then we use the splitting criterion ,http://groupprops.subwiki.org/wiki/Splitting_criterion_for_conjugacy_classes_in_the_alternating_group, as we find a conjugacy class that splits, eg that represented by $(12345)$ in $S_5$.
Is there a way of computing/spotting the two conjugacy classes. One will be $(12345)$ but is there a way of getting the other preferably without doing any calculations.
Looking for general method.