  We analyse the interplay of generalised CP transformations and the
non-Abelian discrete group $T^{\prime}$ and use the semi-direct product $G_f=
T^{\prime}\rtimes H_{\text{CP}}$, as family symmetry acting in the lepton
sector. The family symmetry is shown to be spontaneously broken in a
geometrical manner. In the resulting flavour model, naturally small Majorana
neutrino masses for the light active neutrinos are obtained through the type I
see-saw mechanism. The known masses of the charged leptons, lepton mixing
angles and the two neutrino mass squared differences are reproduced by the
model with a good accuracy. The model allows for two neutrino mass spectra with
normal ordering (NO) and one with inverted ordering (IO). For each of the three
spectra the absolute scale of neutrino masses is predicted with relatively
small uncertainty. The value of the Dirac CP violation (CPV) phase $\delta$ in
the lepton mixing matrix is predicted to be $\delta \cong \pi/2~{\rm or}~
3\pi/2$. Thus, the CP violating effects in neutrino oscillations are predicted
to be maximal (given the values of the neutrino mixing angles) and
experimentally observable. We present also predictions for the sum of the
neutrino masses, for the Majorana CPV phases and for the effective Majorana
mass in neutrinoless double beta decay. The predictions of the model can be
tested in a variety of ongoing and future planned neutrino experiments.
