  Armed with the explicit computation of Schur Multipliers, we offer a
classification of SU(n) orbifolds for n = 2,3,4 which permit the turning on of
discrete torsion. This is in response to the host of activity lately in vogue
on the application of discrete torsion to D-brane orbifold theories. As a
by-product, we find a hitherto unknown class of N = 1 orbifolds with non-cyclic
discrete torsion group. Furthermore, we supplement the status quo ante by
investigating a first example of a non-Abelian orbifold admitting discrete
torsion, namely the ordinary dihedral group as a subgroup of SU(3). A
comparison of the quiver theory thereof with that of its covering group, the
binary dihedral group, without discrete torsion, is also performed.
