The space groups in two dimensions are called wallpaper groups. In three dimensions, the space groups are the symmetry groups
possible in a crystal lattice with the translation symmetry element. There are 230
space groups in 
,
although 11 are mirror images of each other. They
are listed by Hermann-Mauguin symbol in
Cotton (1990).
Space Groups
See also
Crystallographic Point Groups, Hermann-Mauguin Symbol, Lattice Groups, Point Groups, Wallpaper GroupsExplore with Wolfram|Alpha
References
Arfken, G. "Crystallographic Point and Space Groups." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 248-249, 1985.Brown, H.; Blow, R.; Neubser, J.; Wondratschek, H.; and Zassenhaus, H. Crystallographic Groups of Four-Dimensional Space. New York: Wiley, 1978.Buerger, M. J. Elementary Crystallography. New York: Wiley, 1956.Bülow, R.; Neubser, J.; and Wondratschek, H. "On Crystallography in Higher Dimensions. II. Procedure of Computation in
."
Acta Cryst. A 27, 520-523, 1971.Burgbacher, F.; Laemmerzahl,
C.; Macias, A. "Is There a Stable Hydrogen Atom in Higher Dimensions?"
J. Math. Phys. 40, 625-634, 1999.Cotton, F. A. Chemical
Applications of Group Theory, 3rd ed. New York: Wiley, pp. 250-251,
1990.Hermann, C. "Kristallographie in Räumen beliebiger Dimensionszahl.
I. Die Symmetrieoperationen." Acta Cryst. 2, 139-145, 1949.Neubser,
J.; Wondratschek, H.; and Bülow, R. "On Crystallography in Higher Dimensions.
I. General Definitions." Acta Cryst. A 27, 517-520, 1971.Opechowski,
W. Crystallographic
and Metacrystallographic Groups. Amsterdam, Netherlands: North-Holland, 1986.Rabson,
D. A.; Huesman, J. F.; Fisher, B. N. "Cohomology for Anyone."
Found. Phys. 33, 1769-1796, 2003.Schwarzenberger, R. L. E.
"Classification of Crystal Lattices." Proc. Cambridge Philos. Soc. 72,
325-349, 1972.Schwarzenberger, R. L. E. "Crystallography
in Spaces of Arbitrary Dimension." Proc. Cambridge Philos. Soc. 76,
23-32, 1974.Strebel, R. "Burckhardtsche Bestimmung der Raumgruppen
I." Elem. Math. 58, 141-155, 2003.Wondratschek, H.;
Bülow, R.; and Neubser, J. "On Crystallography in Higher Dimensions. III.
Results in 
."
Acta Cryst. A 27, 523-535, 1971.Referenced on Wolfram|Alpha
Space GroupsCite this as:
Weisstein, Eric W. "Space Groups." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SpaceGroups.html