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We use the discrete group \Sigma(81) = (Z_3 x Z_3 x Z_3)\rtimes Z_3 toexplore a particular region of parameter space in the Private Higgs model. Indoing so we suggest a relation among the off-diagonal entries of the neutrinomass matrix and a possible explanation for the muon magnetic moment anomaly,a_\mu^{exp}-a_\mu^{SM} ~ 10^{-9}. We predict three new nearly degenerate Higgsdoublets with masses of order ~ 500 GeV to ~ 1 TeV, and three nearly degenerateSM-singlet TeV-scale neutrinos. The largest scale in the model is ~ 10 TeV, sothere is no severe hierarchy problem. The appendix is devoted to the grouptheory of \Sigma(81).

Lepton Private Higgs and the discrete group \Sigma(81)