 TOPICS # Tetrahedral Group The tetrahedral group is the point group of symmetries of the tetrahedron including the inversion operation. It is one of the 12 non-Abelian groups of order 24. The tetrahedral group has conjugacy classes 1, , , , and (Cotton 1990, pp. 47 and 434). Its multiplication table is illustrated above. The tetrahedral group is implemented in the Wolfram Language as FiniteGroupData["Tetrahedral", "PermutationGroupRepresentation"] and as a point group as FiniteGroupData[ "CrystallographicPointGroup", "Td" , "PermutationGroupRepresentation"].  has a pure rotational subgroup of order 12 denoted (Cotton 1990, pp. 50 and 433). It is isomorphic to the alternating group and has conjugacy classes 1, , , and . It has 10 subgroups: one of length 1, three of length 2, 4 of length 3, one of length 4, and one of length 12. Of these, only the trivial subgroup, subgroup of order 4, and complete subgroup are normal. The pure rotational tetrahedral subgroup is implemented in the Wolfram Language as a point group as FiniteGroupData[ "CrystallographicPointGroup", "T" , "PermutationGroupRepresentation"]. The cycle graph of is illustrated above. The numbers of elements such that for , 2, ..., 12, are 1, 4, 9, 4, 1, 12, 1, 4, 9, 4, 1, 12. Platonic and Archimedean solids that can be generated by group are illustrated above, with the corresponding basis vector summarized in the following table, where is the golden ratio.

 solid basis vector cuboctahedron icosahedron octahedron tetrahedron truncated tetrahedron There is also a point group known as . It has conjugacy classes 1, , , , , , , and (Cotton 1990, pp. 50 and 434). The group is implemented in the Wolfram Language as a point group as FiniteGroupData[ "CrystallographicPointGroup", "Th" , "PermutationGroupRepresentation"].

Finite Group T, Icosahedral Group, Octahedral Group, Point Groups, Polyhedral Group, Tetrahedron

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## References

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, p. 47, 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.C in Applications of Finite Groups. New York: Dover, p. 81, 1987.

## Cite this as:

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