# Finding conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$

Assume $q$ is odd. How does one go about finding the conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$? I know that for $GL_{2}(\mathbb{F}_{q})$, one can consider the possible Jordan Normal Forms of the matrices and with some luck, choose representatives whose conjugacy class is large enough such that when I sum all the conjugacy class sizes I get the whole group.

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$PGL(2,q)$ is just $GL(2,q)$ quotient the center. Thus conjugacy classes in $PGL(2,q)$ are just the equivalence classes of elements in $GL(2,q)$ given by the following relation: $g\equiv h$ if $g=h^kz$, where $g,h,k\in GL(2,q)$ and $z$ is central. Now central elements are multiples of the identity, so all you're doing is taking a normal conjugacy class in $GL(2,q)$, and maybe doing some scalar multiplication. In particular, Jordan normal forms mostly solve the problem. – user641 Mar 1 '12 at 18:53
What do you mean by $h^{k}$ in $g = h^{k}z$? – 387492 Mar 1 '12 at 21:41
The conjugate of $h$ by $k$, $h^k=k^{-1}hk$. – user641 Mar 1 '12 at 22:29