A point group is a group of symmetry operations all of which leave at least one point unmoved. A crystallographic point group is a point group that maps a point lattice onto itself: in three dimensions rotations and rotoinversions are restricted to 1, 2, 3, 4, 6 and , (= m), , , respectively.
Crystallographic point groups occur:
- in vector space, as symmetries of the external shapes of crystals (morphological symmetry), as well as symmetry of the physical
propertiesof the crystal ("vector point group");
- in point space, as site-symmetry groups of points in lattices or in crystal structures, and as symmetries of atomic groups and
coordinationpolyedra ("point point group").
Chapter 10 in International Tables for Crystallography, Volume A