  We perform the first systematic analysis of particle spectra obtained from
heterotic string compactifications on non-Abelian toroidal orbifolds. After
developing a new technique to compute the particle spectrum in the case of
standard embedding based on higher dimensional supersymmetry, we compute the
Hodge numbers for all recently classified 331 non-Abelian orbifold geometries
which yield N=1 supersymmetry for heterotic compactifications. Surprisingly,
most Hodge numbers follow the empiric pattern h^(1,1) - h^(2,1) = 0 mod 6,
which might be related to the number of three standard model generations.
Furthermore, we study the fundamental groups in order to identify the
possibilities for non-local gauge symmetry breaking. Three examples are
discussed in detail: the simplest non-Abelian orbifold S_3 and two more
elaborate examples, T_7 and \Delta(27), which have only one untwisted Kaehler
and no untwisted complex structure modulus. Such models might be especially
interesting in the context of no-scale supergravity. Finally, we briefly
discuss the case of orbifolds with vanishing Euler numbers in the context of
enhanced (spontaneously broken) supersymmetry.
