Crystal symmetry

The spatial transformations that preserve a crystal structure.

The finite orthogonal group obtained by quotienting a crystal space group by its translation subgroup. In three dimensions there are 32 crystallographic point-group types, constrained by preservation of a lattice. It describes macroscopic rotational and reflection symmetry, not the fractional translations of the full space group.

A symmetry combining reflection in a plane with translation parallel to that plane. An $a$ glide translates by half an $a$ period, and a $c$ glide by half a $c$ period. Specifying the reflection trace alone loses the translation and can falsely suggest an ordinary mirror.

A symmetry consisting of rotation about an axis and translation parallel to that axis. A $2_1$ operation combines a half-turn with half a lattice period; applying it twice gives a complete lattice translation. It is a spatial symmetry, distinct from a screw dislocation.