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 Groups## Explore 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 Groups## Cite this as:

Weisstein, Eric W. "Point Groups." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/PointGroups.html