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As is well known, crystals have discrete space translational symmetry. It wasrecently noticed that one-dimensional crystals possibly have discretePoincar\'{e} symmetry, which contains discrete Lorentz and discrete timetranslational symmetry as well. In this paper, we classify the discretePoincar\'{e} groups on two- and three-dimensional Bravais lattices. They arethe candidate symmetry groups of two- or three-dimensional crystals,respectively. The group is determined by an integer generator $g$, and itreduces to the space group of crystals at $g=2$.

Discrete Lorentz symmetry and discrete spacetime translational symmetry in two- and three-dimensional crystals