We analyze the role of the extended Clifford group in classifying the spectra of phase point operators within the framework laid out by [Gibbons et al., Phys. Rev. A 70, 062101 (2004)] for setting up Wigner distributions on discrete phase spaces based on finite fields. To do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field. Auxiliary results include a derivation of the conjugacy classes of .
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