| Summary: | A popular account of the mixing patterns for the three generations of quarks and leptons is through the characters <inline-formula> <math display="inline"> <semantics> <mi>κ</mi> </semantics> </math> </inline-formula> of a finite group <i>G</i>. Here, we introduce a <i>d</i>-dimensional Hilbert space with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mi>c</mi> <mi>c</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, the number of conjugacy classes of <i>G</i>. 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 <i>d</i>-dimensional Pauli group over the characters of these representations. Groups with small <i>d</i> 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.
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