Scotogenic Cobimaximal Dirac Neutrino Mixing from $\mathrm{\Delta }(27)$ and $U(1{)}_{\chi }$

In the context of $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y\times U(1{)}_{\chi }$, where $U(1{)}_{\chi }$ comes from $SO(10)\to SU(5)\times U(1{)}_{\chi }$, supplemented by the non-Abelian discrete $\mathrm{\Delta }(27)$ symmetry for three lepton families, Dirac neutrino masses and their mixing are radiatively generated through dark matter. The gauge $U(1{)}_{\chi }$ symmetry is broken spontaneously. The discrete $\mathrm{\Delta }(27)$ 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 $Z_2$, $Z_3$, or $U(1{)}_D$ depending on what scalar breaks $U(1{)}_{\chi }$. Cobimaximal neutrino mixing, i.e. ${\theta }_{13}\mathrm{\ne }0$, ${\theta }_{23}=\pi \mathrm{/}4$, and ${\delta }_{CP}=\pm \pi \mathrm{/}2$, may also be obtained.