Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
List of spherical symmetry groups
open-in-new
Selected <a href="/facts/Point_groups_in_three_dimensions/Uc4MRIyl">point groups in three dimensions</a><table><tbody><tr><td>Involutional symmetryCs, (*)[ ] = </td><td><a href="/facts/Cyclic_symmetry_in_three_dimensions/wQUCezbQ">Cyclic symmetry</a>Cnv, (*nn)[n] = </td><td><a href="/facts/Dihedral_symmetry_in_three_dimensions/xNRpsIAF">Dihedral symmetry</a>Dnh, (*n22)[n,2] = </td></tr><tr><th colspan="4"><a href="/facts/Polyhedral_group/ynwsQLXE">Polyhedral group</a>, [n,3], (*n32)</th></tr><tr><td><a href="/facts/Tetrahedral_symmetry/hIVlf9pu">Tetrahedral symmetry</a>Td, (*332)[3,3] = </td><td><a href="/facts/Octahedral_symmetry/qLqhYxgm">Octahedral symmetry</a>Oh, (*432)[4,3] = </td><td><a href="/facts/Icosahedral_symmetry/pDlwAy23">Icosahedral symmetry</a>Ih, (*532)[5,3] = </td></tr></tbody></table> <p>Finite spherical symmetry groups are also called <a href="/facts/Point_groups_in_three_dimensions/Uc4MRIyl">point groups in three dimensions</a>. There are five fundamental symmetry classes which have triangular fundamental domains: <a href="/facts/Dihedral_symmetry/1PUlRh4M">dihedral</a>, <a href="/facts/Cyclic_symmetry/JL5zfFuL">cyclic</a>, <a href="/facts/Tetrahedral_symmetry/hIVlf9pu">tetrahedral</a>, <a href="/facts/Octahedral_symmetry/qLqhYxgm">octahedral</a>, and <a href="/facts/Icosahedral_symmetry/pDlwAy23">icosahedral</a> symmetry. </p><p>This article lists the groups by <a href="/facts/Schoenflies_notation/c4y0GtEt">Schoenflies notation</a>, <a href="/facts/Coxeter_notation/ArUw6goe">Coxeter notation</a>, <a href="/facts/Orbifold_notation/HCbBoa2J">orbifold notation</a>, and order. <a href="/facts/John_Horton_Conway/g3PA1zoK">John Conway</a> uses a variation of the Schoenflies notation, based on the groups' <a href="/facts/Quaternion/T3tnu7mX">quaternion</a> 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 <a href="/facts/Central_inversion/UzswAm8N">central inversion</a>. </p><p><a href="/facts/Hermann%25E2%2580%2593Mauguin_notation/NGHDFXKC">Hermann–Mauguin notation</a> (International notation) is also given. The <a href="/facts/Crystallography/9B5toeqG">crystallography</a> groups, 32 in total, are a subset with element orders 2, 3, 4 and 6. </p>