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The generalized CP transformations can only be consistently defined in thecontext of $\Delta(3n^2)$ lepton symmetry if a certain subset of irreduciblerepresentations are present in a model. We perform a comprehensive analysis ofthe possible automorphisms and the corresponding CP transformations of the$\Delta(3n^2)$ group. It is sufficient to only consider three automorphisms if$n$ is not divisible by 3 while additional eight types of CP transformationscould be imposed for the case of $n$ divisible by 3. We study the lepton mixingpatterns which can be derived from the $\Delta(3n^2)$ family symmetry andgeneralized CP in the semi-direct approach. The PMNS matrix is determined to bethe trimaximal pattern for all the possible CP transformations, and it can onlytake two distinct forms.

Generalized CP and $\Delta (3n^2)$ Family Symmetry for Semi-Direct Predictions of the PMNS Matrix