Point Groups

The symmetry groups possible in a crystal lattice without the translation symmetry element. Although an isolated object may have an arbitrary Schönflies Symbol, the requirement that symmetry be present in a lattice requires that only 1, 2, 3, and 6-fold symmetry axes are possible (the Crystallography Restriction), which restricts the number of possible point groups to 32: $C_i$, $C_s$, $C_1$, $C_2$, $C_3$, $C_4$, $C_6$, $C_{2h}$, $C_{3h}$, $C_{4h}$, $C_{6h}$, $C_{2v}$, $C_{3v}$, $C_{4v}$, $C_{6v}$, $D_2$, $D_3$, $D_4$, $D_6$ (the Dihedral Groups), $D_{2h}$, $D_{3h}$, $D_{4h}$, $D_{6h}$, $D_{2d}$, $D_{3d}$, $O$, $O_h$ (the Octahedral Group), $S_4$, $S_6$, $T$, $T_h$, and $T_d$ (the Tetrahedral Group).

See also Crystallography Restriction, Dihedral Group, Group, Group Theory, Hermann-Mauguin Symbol, Lattice Groups, Octahedral Group, Schönflies Symbol, Space Groups, Tetrahedral Group

References

Arfken, G. ``Crystallographic Point and Space Groups.'' Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 248-249, 1985.

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, p. 379, 1990.

Lomont, J. S. ``Crystallographic Point Groups.'' §4.4 in Applications of Finite Groups. New York: Dover, pp. 132-146, 1993.