  We analyse the role of the Extended Clifford group in classifying the spectra
of phase point operators within the framework laid out by Gibbons et al 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 ${\rm ESL}(2, \mathbb{F}_N)$.
