  We study abelian and non-abelian orbifolds of the ABJM model. We compute the
precise moduli space of these models by analyzing the classical BPS equations
for the theory on the cylinder, which include classical solutions of magnetic
monopole operators. These determine the chiral ring of the theory, and thus
they provide the complete set of order parameters determining the classical
vacua of the theory. We show that the proper quantization of these
semiclassical solutions gives us the topology of moduli space, including the
additional quotient information due to the Chern-Simons levels. In general, we
find that in the dual M-theory setup, the M-theory fiber is divided by the
product of the Chern-Simons level times the order of the orbifold group, even
in the non-abelian case. This depends non-trivially on how the different
Chern-Simons terms have different levels in these constructions. We also see a
direct relation in this setup between the Chern-Simons levels of the different
groups and fluxes for fractional brane cycles. We also show that the problem of
the moduli space can be much more easily analyzed by using the method of images
and representation theory of crossed product algebras rather than dealing only
with the quiver theory data.
