Rotation Groups
The rotation groups, i.e. the finite subgroups of SO(3), are: the cyclic groups Cn (the rotation group of a regular pyramid), the dihedral groups Dn (the rotation group of a regular prism, or regular bipyramid), and the rotation groups T, O and I of a regular tetrahedron, octahedron/cube and icosahedron/dodecahedron.
In particular, the dihedral groups D3, D4 etc. are the rotation groups of plane regular polygons embedded in three-dimensional space, and such a figure may be considered as a degenerate regular prism. Therefore it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group.
- An object with symmetry group Cn, Cnh, Cnv or S2n has rotation group Cn.
- An object with symmetry group Dn, Dnh, or Dnd has rotation group Dn.
- An object with one of the other seven symmetry groups has as rotation group the corresponding one without subscript: T, O or I.
The rotation group of an object is equal to its full symmetry group if and only if the object is chiral. In other words, the chiral objects are those with their symmetry group in the list of rotation groups.
Given in Schönflies notation, Coxeter notation, (orbifold notation), the rotation subgroups are:
| Nonrotation group | Rotational subgroup | ||
|---|---|---|---|
| Reflectional | Reflection/rotational | Glide reflectional | |
| Cnv, (*nn) | Cnh, (n*) | S2n, (nx) | Cn, +, (nn) |
| Dnh, (*n22) | Dnd, (2*n) | Dn, +, (n22) | |
| Td, (*332) | Th, (3*2) | T, +, (332) | |
| Oh, (*432) | O, +, (432) | ||
| Ih, (*532) | I, +, (532) | ||
Read more about this topic: Point Groups In Three Dimensions
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