A point group is a group of symmetry operations which all leave at least one point unmoved. 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, 4, and 6-fold symmetry axes are possible (the crystallography restriction), which restricts the number of possible so-called crystallographic point groups to 32.
Point Groups
See also
Crystallographic Point Groups, Crystallography Restriction, Cubic Group, Cyclic Group, Dihedral Group, Octahedral Group, Schönflies Symbol, Space Groups, Tetrahedral Group, Wallpaper GroupsExplore with Wolfram|Alpha
References
Cotton, F. A. "Character Tables for the Chemically Important Symmetry Groups." Appendix IIA in Chemical Applications of Group Theory, 3rd ed. New York: Wiley, pp. 426-436, 1990.Hahn, T. (Ed.). International Tables for Crystallography, Vol. A: Space Group Symmetry, 4th ed. Dordrecht, Netherlands: Kluwer, p. 752, 1995.Veysseyre, R. and Veysseyre H. "Point Groups of Five-Dimensional Space." Acta Cryst. A 58, 429-433, 2002.Referenced on Wolfram|Alpha
Point GroupsCite this as:
Weisstein, Eric W. "Point Groups." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PointGroups.html