List of finite spherical symmetry groups

From Wikipedia, the free encyclopedia

Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.

Table info: Polyhedral group, [n,3], (*n32)...
Selected point groups in three dimensions
Involutional symmetry
Cs, (*)
[ ] = CDel_node_c2.png
Cyclic symmetry
Cnv, (*nn)
[n] = CDel_node_c1.pngCDel_n.pngCDel_node_c1.png
Dihedral symmetry
Dnh, (*n22)
[n,2] = CDel_node_c1.pngCDel_n.pngCDel_node_c1.pngCDel_2.pngCDel_node_c1.png
Polyhedral group, [n,3], (*n32)
Tetrahedral symmetry
Td, (*332)
[3,3] = CDel_node_c1.pngCDel_3.pngCDel_node_c1.pngCDel_3.pngCDel_node_c1.png
Octahedral symmetry
Oh, (*432)
[4,3] = CDel_node_c2.pngCDel_4.pngCDel_node_c1.pngCDel_3.pngCDel_node_c1.png
Icosahedral symmetry
Ih, (*532)
[5,3] = CDel_node_c2.pngCDel_5.pngCDel_node_c2.pngCDel_3.pngCDel_node_c2.png

This article lists the groups by Schoenflies notation, Coxeter notation,[1] orbifold notation,[2] and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion.[3]

Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6.[4]