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Icosahedral Group


IcosahedralGroupIhTable

The icosahedral group I_h is the group of symmetries of the icosahedron and dodecahedron having order 120, equivalent to the group direct product A_5×Z_2 of the alternating group A_5 and cyclic group Z_2. The icosahedral group consists of the conjugacy classes 1, 12C_5, 12C_5^2, 20C_3, 15C_2, i, 12S_(10), 12S_(10)^3, 20S_6, and 15sigma (Cotton 1990, pp. 49 and 436). Its multiplication table is illustrated above. The icosahedral group is a subgroup of the special orthogonal group SO(3). The icosahedal group I_h is implemented in the Wolfram Language as FiniteGroupData["Icosahedral", "PermutationGroupRepresentation"].

Icosahedral symmetry is possible as a rotational group but is not compatible with translational symmetry. As a result, there are no crystals with this symmetry and so, unlike the octahedral group O_h and tetrahedral group T_h, I_h is not one of the 32 point groups.

IcosahedralGroupIhPolyhedra

The great rhombicosidodecahedron can be generated using the matrix representation of I_h using the basis vector (phi,3,2phi), where phi is the golden ratio.

IcosahedralGroupITable

The icosahedral group I_h has a pure rotation subgroup denoted I that is isomorphic to the alternating group A_5. I is of order 60 and has conjugacy classes 1, 12C_5, 12C_5^2, 20C_3, and 15C_2 (Cotton 1990, pp. 50 and 436). Like I_h, I is not a point group. Its multiplication table is illustrated above. The group I is currently not implemented as a separate group in the Wolfram Language.

IcosahedralGroupIPolyhedra

Platonic and Archimedean solids that can be generated by group I are illustrated above, with the corresponding basis vector summarized in the following table, where phi is the golden ratio and a and b are the largest positive roots of two sixth-order polynomials.


See also

Alternating Group, Bipolyhedral Group, Dodecahedron, Icosahedron, Octahedral Group, Point Groups, Polyhedral Group, Special Orthogonal Group, Tetrahedral Group

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References

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, pp. 48-50, 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. "Icosahedral Group." §3.10.E in Applications of Finite Groups. New York: Dover, p. 82, 1987.

Referenced on Wolfram|Alpha

Icosahedral Group

Cite this as:

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

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