If $q$ is a prime power, then the conjugacy classes in $GL(n,q)$ are determined completely by minimal polynomials.
Now in the subgroup $SL(n,q)$, these conjugacy classes can split, in the sense they form multiple classes within $SL(n,q)$.
I remember reading in some paper (now long lost) that one can also determine representatives for each of these new conjugacy classes from the minimal polynomial. It was something along the lines of looking at the repeated factors in the factorization of the minimal polynomial.
Can someone remind me what this criterion is? Thanks!