Size of conjugacy classes in SL(2,3) I've been given the representations of the conjugacy classes for a group presentation $G = <x,y,z | x^2 = y^3 = z^3 = xyz>$ which is isomorphic to $SL(2,\mathbb{F}_3)$ which are:
$\bigl(\begin{smallmatrix}
1&0\\ 0&1
\end{smallmatrix} \bigr)$ $\bigl(\begin{smallmatrix}
2&0\\ 0&2
\end{smallmatrix} \bigr)$ $\bigl(\begin{smallmatrix}
0&2\\ 1&0
\end{smallmatrix} \bigr)$ $\bigl(\begin{smallmatrix}
1&1\\ 1&0
\end{smallmatrix} \bigr)$ $\bigl(\begin{smallmatrix}
1&0\\ 2&1
\end{smallmatrix} \bigr)$ $\bigl(\begin{smallmatrix}
2&0\\ 1&2
\end{smallmatrix} \bigr)$ $\bigl(\begin{smallmatrix}
2&0\\ 2&2
\end{smallmatrix} \bigr)$
Now this group has order 24, and rather than write out all of the calculations for the conjugacy class by hand, I was wondering whether there was a more subtle way to calculate the sizes of the conjugacy classes? I began by trying to multiply a conjugacy class representative by $\bigl(\begin{smallmatrix}
a&b\\ c&d
\end{smallmatrix} \bigr) \in SL(2,\mathbb{F}_3)$ and see if I get a matrix of a certain form but it didn't seem to help much, if at all. 
 A: Here is one thing that will help.  Note if $z \in Z(G)$, the center of $G$, then
$$ z x^{-1}g x =  x^{-1} zg x, \quad x, g \in G.$$
This means that multiplication by an element of the center gives an action on the set of conjugacy classes, i.e., if $C$ is a conjugacy class in $G$, so is $zC$.
In this case, the center is simply $Z=\{ \pm I \}$.  In fact $SL(2,3)/Z = PSL(2,3) \simeq A_4$ (cf.\ Deitrich's comment), so this reduces the problem essentially (in principle) to determining the conjugacy classes of $A_4$.  From the representatives (not representations) you have, there are obviously 2 pairs that just differ by $-I = 2I$.
Now the standard way to compute conjugacy classes (which maybe you know) is for each representative $g$, compute the centralizer
$$ C_G(g) = \{ x \in G : xgx^{-1} = g \}.$$
This is a subgroup, and the elements of the conjugacy class $g$ are given by $ygy^{-1}$ where $y$ runs over a set of coset representatives $G/C_G(g)$ (in you're not familiar with this, think about why).  So if you just want the sizes of the conjugacy classes (namely $|G|/|C_G(g)|$), you just need to compute the order of each $C_G(g)$, 
This still requires some computation, but it shouldn't be too bad.  (The case of the identity is trivial (what is the centralizer?), so then you're basically left with 4 conjugacy classes to consider.)
