1. The problem statement, all variables and given/known data e) If H ∼= Z3 × Z3 show that there are exactly 2 conjugacy classes of elements of order 2 in Aut(Z3 × Z3) = GL(2, Z3). f) Choosing an element of each conjugacy class in e), construct two semidirect products of H and K. By counting orders of elements in each such group, show that the two groups you construct are not isomorphic. 2. Relevant equations 3. The attempt at a solution So far I just have thoughts really... As for finding the two conjugacy classes of GL(2,Z3) it seems that one of these conjugacy classes must be the identity matrix, because the identity matrix conjugated by anything will just return itself. So does one of the two conjugacy classes have the identity matrix and the other conjugacy class have everything else? Wouldn't these conjugacy classes have equal number of elements in each of them though? I am quite confused. I am quite stuck. If anybody could shed some light on this topic for me that'd be awesome, I'll be editing this post as I think of new ideas.