List of symmetry groups

Coxeter groups
A special type of symmetry group, these are the symmetries of Wythoffian polytopes. Those that can be found in Euclidean tilings are named after Lie groups related to them, while the others are given names to continue the alphabetical theme.


 * An: n-dimensional simplex (o3o3o3o...o)
 * Bn: n-dimensional cube/orthoplex (o4o3o3o...o), also called Cn or BCn (both Lie groups Bn and Cn produce the same Coxeter group)
 * Dn: n-dimensional demicube (o3o3o...o *b3o)
 * En: for 6≤n≤8, Gosset polytopes (o3o3o...o *c3o)
 * F4: the 24-cell (o3o4o3o)
 * G2: the hexagon (o6o) (redundant with I2(6) but given a special name due to being used in tilings)
 * Hn: for 2≤n≤4, pentagonal polytopes (o5o3o3o...o)
 * I2(p): the p-gon (opo)

0D
The only isometry of 0D space is identity.


 * Pointic (A0), order 1

1D
1D is the first dimension where symmetrical and asymmetrical objects are distinguished. Reflections and translations are isometries, but translations in general are not useful for finite polytopes.


 * Dyadic (A1) a single reflection
 * I|Monic (I) completely asymmetrical

2D
An infinite number of possible degrees of rotation occur, by composing two reflections.


 * n-fold reflection symmetry (I2(n)), can be rotated 1/n turns or reflected, forming n lines of reflection
 * Reflection (A1×I) a single reflection possible
 * Rectangular (K2) 2 reflections or a 1/2-turn rotation
 * Triangular (A2) 3 reflections or a 1/3-turn rotation
 * Square (B2) 4 reflections or a 1/4-turn rotation
 * Pentagonal (H2) 5 reflections or a 1/5-turn rotation
 * Hexagonal (G2) 6 reflections or a 1/6-turn rotation
 * n-fold rotation symmetry (I2(n)+), can be rotated 1/n turns but no reflections
 * Identity (I×I) no symmetries except identity
 * Inversion (K2+) can be inverted (same as 1/2-turn rotation) but nothing else
 * Chiral triangular (A2+) can be turned 1/3 turns
 * Chiral square (B2+) can be turned 1/4 turns
 * Chiral pentagonal (H2+) can be turned 1/5 turns
 * Chiral hexagonal (G2+) can be turned 1/6 turns
 * Chiral hexagonal (G2+) can be turned 1/6 turns

3D
Prismatic, pyramidal, antiprismatic (and variants), tetrahedral (chiral), cubic (chiral, pyritohedral), dodecahedral (chiral)

4D
Pentachoric (chiral, decachoric, chiral decachoric, ionic decachoric), tesseractic (chiral, pyritotesseractic), demitesseractic (chiral), icositetrachoric (chiral, tetracontoctachoric, chiral tetracontoctachoric, ionic tetracontoctachoric, pyritoicositetrachoric, toxitic, oxitic), hecatonicosachoric (chiral, ixitic, ixoic), prismatic and antiprismatic, duoprismatic, swirlchoric, gyrochoric