  In this work we introduce a symmetry classification for electronic density
waves which break translational symmetry due to commensurate wave vector
modulations. The symmetry classification builds on the concept of extended
point groups: symmetry groups which contain, in addition to the lattice point
group, translations that do not map the enlarged unit cell of the density wave
to itself, and become "non-symmorphic"-like elements. Multi-dimensional
representations of the extended point group are associated with degenerate wave
vectors. Electronic properties such as (nodal) band degeneracies and
topological character can be straightforwardly addressed, and often follow
directly. To further flesh out the idea of symmetry, the classification is
constructed so as to manifestly distinguish time-reversal invariant charge
(i.e., site and bond) order, and time-reversal breaking flux order. For the
purpose of the present work we particularize to spin-rotation invariant density
waves. As a first example of the application of the classification we consider
the density waves of a simple single- and two-orbital square lattice model. The
main objective, however, is to apply the classification to two-dimensional
($2D$) hexagonal lattices, specifically the triangular and the honeycomb
lattices. The multi-component density waves corresponding to the commensurate
$M$-point ordering vectors are worked out in detail. To show that our results
generally apply to $2D$ hexagonal lattices, we develop a general low-energy
$SU(3)$ theory of (spinless) saddle point electrons.
