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 octahedral point groups are also crystallographic point groups, the icosahedral group 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 Explore with Wolfram|Alpha References Arfken, G. "Crystallographic Point and Space Groups." Mathematical Methods for Physicists, 3rd ed. Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. Hahn, T. (Ed.). International Tables for Crystallography, Vol. A: Space Group Symmetry, 4th ed. Lomont, J. S. "Crystallographic Point Groups." §4.4 in Applications of Finite Groups. 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. Referenced on Wolfram|Alpha Crystallographic Point Groups Cite this as: Weisstein, Eric W. "Crystallographic Point Groups." From MathWorld https://mathworld.wolfram.com/CrystallographicPointGroups.html
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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. 