Conjugacy and the Class Equation · Superchallenge problem

Conjugate matrices always have the same determinant (since $\det(XMX^{-1})=\det(M)$), so each of $GL(2,3)$'s $8$ conjugacy classes lies entirely inside one determinant value. Of these $8$ classes, how many consist of matrices with determinant $1$ (i.e. lie in $SL(2,3)$)?