  We present a complete intersection Calabi-Yau manifold Y that has Euler
number -72 and which admits free actions by two groups of automorphisms of
order 12. These are the cyclic group Z_12 and the non-Abelian dicyclic group
Dic_3. The quotient manifolds have chi=-6 and Hodge numbers (h^11,h^21)=(1,4).
With the standard embedding of the spin connection in the gauge group, Y gives
rise to an E_6 gauge theory with 3 chiral generations of particles. The gauge
group may be broken further by means of the Hosotani mechanism combined with
continuous deformation of the background gauge field. For the non-Abelian
quotient we obtain a model with 3 generations with the gauge group broken to
that of the standard model. Moreover there is a limit in which the quotients
develop 3 conifold points. These singularities may be resolved simultaneously
to give another manifold with (h^11,h^21)=(2,2) that lies right at the tip of
the distribution of Calabi-Yau manifolds. This strongly suggests that there is
a heterotic vacuum for this manifold that derives from the 3 generation model
on the quotient of Y. The manifold Y may also be realised as a hypersurface in
the toric variety. The symmetry group does not act torically, nevertheless we
are able to identify the mirror of the quotient manifold by adapting the
construction of Batyrev.
