Generalised CP and $\Delta (6n^2)$ Family Symmetry in Semi-Direct Models of Leptons

We perform a detailed analysis of $\Delta (6n^2)$ family symmetry combined with a generalised CP symmetry in the lepton sector, breaking to different remnant symmetries $G_{\nu}$ in the neutrino and $G_{l}$ in the charged lepton sector, together with different remnant CP symmetries in each sector. We discuss the resulting mass and mixing predictions for $G_{\nu}=Z_2$ with $G_{l}=K_4,Z_p,p>2$ and $G_{\nu}=K_4$ with $G_{l}=Z_2$. All cases correspond to the preserved symmetry smaller than the full Klein symmetry, as in the semi-direct approach, leading to predictions which depend on a single undetermined real parameter, which mainly determines the reactor angle. We focus on five phenomenologically allowed cases for which we present the resulting predictions for the PMNS parameters as a function of $n$, as well as the predictions for neutrinoless double beta decay.