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We analyse the role of the Extended Clifford group in classifying the spectraof phase point operators within the framework laid out by Gibbons et al forsetting up Wigner distributions on discrete phase spaces based on finitefields. To do so we regard the set of all the discrete phase spaces as asymplectic vector space over the finite field. Auxiliary results include aderivation of the conjugacy classes of ${\rm ESL}(2, \mathbb{F}_N)$.

Spectra of phase point operators in odd prime dimensions and the extended Clifford group