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Octahedral Group


OctahedralGroupOhTable

O_h is the point group of symmetries of the octahedron having order 48 that includes inversion. It is also the symmetry group of the cube, cuboctahedron, and truncated octahedron. It has conjugacy classes 1, 8C_3, 6C_4, 6C_2, 3C_2=C_4^2, i, 6S_4, 8S_6, 3sigma_h, and 6sigma_4 (Cotton 1990). Its multiplication table is illustrated above. The octahedral group O_h is implemented in the Wolfram Language as FiniteGroupData["Octahedral", "PermutationGroupRepresentation"] and as a point group as FiniteGroupData[{"CrystallographicPointGroup", "Oh"}, "PermutationGroupRepresentation"].

OctahedralGroupOhPolyhedra

The great rhombicuboctahedron can be generated using the matrix representation of O_h using the basis vector (1,3-sqrt(2),sqrt(2)-1).

OctahedralGroupOTable

The octahedral group O_h has a pure rotation subgroup denoted O that is isomorphic to the tetrahedral group T_d. O is of order 24 and has conjugacy classes 1, 8C_3, 3C_2, 6C_4, and 6C_2 (Cotton 1990, pp. 50 and 434). Its multiplication table is illustrated above. The pure rotational octahedral subgroup O is implemented in the Wolfram Language as a point group as FiniteGroupData[{"CrystallographicPointGroup", "O"}, "PermutationGroupRepresentation"].

CycleGraphO

The cycle graph of O is illustrated above.

OctahedralGroupOPolyhedra

Platonic and Archimedean solids that can be generated by group O are illustrated above, with the corresponding basis vector summarized in the following table, where a and b are the largest positive roots of the cubic polynomials a^3+a^2+a-1=0 and b^3+b^2-3b-1=0.


See also

Cube, Cuboctahedron, Icosahedral Group, Octahedron, Point Groups, Polyhedral Group, Tetrahedral Group, Truncated Octahedron

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References

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, pp. 47-49, 1990.Coxeter, H. S. M. "The Polyhedral Groups." §3.5 in Regular Polytopes, 3rd ed. New York: Dover, pp. 46-47, 1973.Lomont, J. S. "Octahedral Group." §3.10.D in Applications of Finite Groups. New York: Dover, p. 81, 1987.

Referenced on Wolfram|Alpha

Octahedral Group

Cite this as:

Weisstein, Eric W. "Octahedral Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OctahedralGroup.html

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