I thought I had the hang of algebra but the questions that I am struggling through at the moment are raising so many problems!

This one has been driving me mad:

We are given the fact that if G is a group of order 24, generated by elements x, y, z such that

then G is isomorphic to SL

2(Z

3) with the isomorphism given by:

x mapped to the 2x2 matrix (0,2;1,0) (i.e 0 2 on the top row and 1 0 on the bottom), y is mapped to the marix (2,0 ; 1,2) and z is mapped to (0,1 ; 2,1). I do not have to prove this and can use it throughout the question.

I have to compute the conjugacy classes, the centraliser of an element in each conjugacy class and the centre of G.

The hint given says to start by conjugating a generic matrix (a,b ; c,d) by each generator. I have done this but am now stuck. I know I need to find the elements in the group but I dont want a 24 x 24 table and am not sure how to reduce it by much. Also, am I meant to be working with matrices or the x,y and z? I tried conjugating generators by generators but the inverses of the matrices leave me with matrices with entries like 1/2 or 1/4 and these are not in Z

3!

This is an assessed question so hints would be great or perhaps a similar example. On top of all this I'm not even sure how to tell the difference between the centraliser of an element and the centre of G!

Thanks!