  A popular account of the mixing patterns for the three generations of quarks
and leptons is through the characters $\kappa$ of a finite group $G$. Here we
introduce a $d$-dimensional Hilbert space with $d=cc(G)$, the number of
conjugacy classes of $G$. Groups under consideration should follow two rules,
(a) the character table contains both two- and three-dimensional
representations with at least one of them faithful and (b) there are minimal
informationally complete measurements under the action of a $d$-dimensional
Pauli group over the characters of these representations. Groups with small $d$
that satisfy these rules coincide in a large part with viable ones derived so
far for reproducing simultaneously the CKM (quark) and PNMS (lepton) mixing
matrices. Groups leading to physical $CP$ violation are singled out.
