  Nematic phases, breaking spontaneously rotational symmetry, provide for
ubiquitously observed states of matter in both classical and quantum systems.
These nematic states may be further classified by their $N$--fold rotational
invariance described by cyclic groups $C_N$ in 2+1D. Starting from the space
groups of underlying $2d$ crystals, we present a general classification scheme
incorporating $C_N$ nematic phases that arise from dislocation-mediated melting
and discuss the conventional tensor order parameters. By coupling the $O(2)$
matter fields to the $Z_N$ lattice gauge theory, an unified $O(2)/Z_N$ lattice
gauge theory is constructed in order to describe all these nematic phases. This
lattice gauge theory is shown to reproduce the $C_N$ nematic-isotropic liquid
phase transitions and contains an additional deconfined phase. Finally, using
our $O(2)/Z_N$ gauge theory framework, we discuss phase transitions between
different $C_N$ nematics.
