In the context of S U (3) C×S U (2) L×U (1) Y×U (1) χ , where U (1) χ comes from S O (10 ) →S U (5 ) ×U (1) χ , supplemented by the non-Abelian discrete ∆ (27 ) symmetry for three lepton families, Dirac neutrino masses and their mixing are radiatively generated through dark matter. The gauge U (1) χ symmetry is broken spontaneously. The discrete ∆ (27 ) symmetry is broken softly and spontaneously. Together, they result in two residual symmetries, a global U (1) L lepton number and a dark symmetry, which may be Z2, Z3, or U (1) D depending on what scalar breaks U (1) χ . Cobimaximal neutrino mixing, i.e. θ13≠0 , θ23=π /4 , and δCP=±π /2 , may also be obtained.