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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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    The conjugate of $h$ by k, $h^k=k^{-1}hk$.2012-03-01

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