  We discuss the possibility of enforcing a massless Majorana neutrino in the
direct and semi-direct approaches to lepton mixing, in which the PMNS matrix is
partly predicted by subgroups of a discrete family symmetry, extending previous
group searches up to order 1535. We find a phenomenologically viable scheme for
the semi-direct approach based on $Q(648)$ which contains $\Delta(27)$ and the
quaternion group as subgroups. This leads to novel predictions for the first
column of the PMNS matrix corresponding to a normal neutrino mass hierarchy
with $m_1=0$, and sum rules for the mixing angles and phase which are
characterised by the solar angle being on the low side $\theta_{12}\sim
31^{\circ}$ and the Dirac (oscillation) CP phase $\delta$ being either about
$\pm 45^\circ$ or $\pm \pi$.
