Crystallographic Point Groups

The crystallographic point groups are the point groups in which translational periodicity is required (the so-called crystallography restriction). There are 32 such groups, summarized in the following table which organizes them by Schönflies symbol type.

typepoint groups
nonaxialC_i, C_s
cyclicC_1, C_2, C_3, C_4, C_6
cyclic with horizontal planesC_(2h), C_(3h), C_(4h), C_(6h)
cyclic with vertical planesC_(2v), C_(3v), C_(4v), C_(6v)
dihedralD_2, D_3, D_4, D_6
dihedral with horizontal planesD_(2h), D_(3h), D_(4h), D_(6h)
dihedral with planes between axesD_(2d), D_(3d)
improper rotationS_4, S_6
cubic groupsT, T_h, T_d, O, O_h

Note that while the tetrahedral T_d and octahedral O_h point groups are also crystallographic point groups, the icosahedral group I_h is not. The orders, classes, and group operations for these groups can be concisely summarized in their character tables.

See also

Character Table, Crystallography Restriction, Dihedral Group, Group, Group Theory, Hermann-Mauguin Symbol, Lattice Groups, Octahedral Group, Point Groups, Schönflies Symbol, Space Groups, Tetrahedral Group, Wallpaper Groups

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Arfken, G. "Crystallographic Point and Space Groups." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 248-249, 1985.Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, p. 379, 1990.Hahn, T. (Ed.). International Tables for Crystallography, Vol. A: Space Group Symmetry, 4th ed. Dordrecht, Netherlands: Kluwer, p. 752, 1995.Lomont, J. S. "Crystallographic Point Groups." §4.4 in Applications of Finite Groups. New York: Dover, pp. 132-146, 1993.Souvignier, B. "Enantiomorphism of Crystallographic Groups in Higher Dimensions with Results in Dimensions Up to 6." Acta Cryst. A 59, 210-220, 2003.Yale, P. B. "Crystallographic Point Groups." §3.4 in Geometry and Symmetry. New York: Dover, pp. 103-108, 1988.

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Crystallographic Point Groups

Cite this as:

Weisstein, Eric W. "Crystallographic Point Groups." From MathWorld--A Wolfram Web Resource.

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