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.

type point groups nonaxial , cyclic ,
,
,
,
cyclic with horizontal planes ,
,
,
cyclic with vertical planes , , , dihedral , , , dihedral with horizontal
planes , , , dihedral with
planes between axes , improper rotation , cubic groups , , , ,

Note that while the tetrahedral and octahedral point groups are also crystallographic point groups,
the icosahedral group 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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References 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. Referenced
on Wolfram|Alpha Crystallographic Point Groups
Cite this as:
Weisstein, Eric W. "Crystallographic Point
Groups." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/CrystallographicPointGroups.html

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