O E 8 C3 3 C2 6 C4 6 C2' <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1 1 1 1 1 ... ... ..... ....... T........ ........... T............ A2 1 1 1 -1 -1 ... ... ..... T...... ......... ........... .T........... E 2 -1 2 0 0 ... ... TT... ....... .TT...... TT......... ..TT......... T1 3 0 -1 1 -1 TTT TTT ..... ....TTT ...TTT... ..TTTTTT... ....TTT...... T2 3 0 -1 -1 1 ... ... ..TTT .TTT... ......TTT ........TTT .......TTTTTT Symmetry of Rotations and Cartesian products A1 g+i+k+l+m x^{4}+y^{4}+z^{4}, x^{2}y^{2}z^{2} A2 f+i+j+l+m xyz, x^{4}(y^{2}−z^{2})+y^{4}(z^{2}−x^{2})+z^{4}(x^{2}−y^{2}) E d+g+h+i+j+2k+l+2m {x^{2}−y^{2}, 2z^{2}−x^{2}−y^{2}}, {x^{4}−y^{4}, 2z^{4}−x^{4}−y^{4}}, {xyz(x^{2}−y^{2}), xyz(2z^{2}−x^{2}−y^{2})}, {2z^{6}−x^{6}−y^{6}, x^{6}−y^{6}} T1 R+p+f+g+2h+i+2j+2k+3l+2m {R_{x}, R_{y}, R_{z}}, {x, y, z}, {x^{3}, y^{3}, z^{3}}, {xy(x^{2}−y^{2}), xz(x^{2}−z^{2}), yz(y^{2}−z^{2})}, {x^{2}y^{2}z, x^{2}yz^{2}, xy^{2}z^{2}}, {x^{5}, y^{5}, z^{5}}, {xy(x^{4}−y^{4}), xz(x^{4}−z^{4}), yz(y^{4}−z^{4})} T2 d+f+g+h+2i+2j+2k+2l+3m {xy, xz, yz}, {x(z^{2}−y^{2}), y(z^{2}−x^{2}), z(x^{2}−y^{2})}, {x^{2}yz, xy^{2}z, xyz^{2}}, {x(x^{4}−z^{4}), y(x^{4}−z^{4}), z(x^{4}−y^{4})}, {x^{4}yz, xy^{4}z, xyz^{4}}, {x^{3}y^{3}, x^{3}z^{3}, y^{3}z^{3}} Notes: α The order of the O point group is 24, and the order of the principal axis (C_{4}) is 4. The group has 5 irreducible representations. β The O point group is isomorphic to T_{d}. It is also isomorphic to the Symmetric Group Sym(4). γ The O point group is generated by two symmetry elements, which can be chosen as two distinct C_{4} axes, or any C_{4} with any C_{3}, or any C_{4} (or C_{3}) with a non-orthogonal C_{2}^{′}. δ The lowest nonvanishing multipole moment in O is 16 (hexadecapole moment). ε This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. ζ The point group is chiral, as it does not contain any mirroring operation. η This point group has several symmetry elements of order 3 or higher which are not coaxial. Therefore, it has at least three-dimensional irreducible representations. The symmetry described by this group may be called “isometric”, as the three Cartesian directions are degenerate. θ This point group corresponds to cubic symmetry, because it is isometric but has no C_{5} axis. More precisely, it is octahedral because it has four-fold axes of rotation. Note that the form of the Cartesian products and their ordering in the table above are somewhat arbitrary. ι All characters are integers because the order of the principal axis is 1,2,3,4 or 6. Such point groups are also referred to as “crystallographic point groups”, as they are compatible with periodic lattice symmetry. There are exactly 32 such groups: C_{1},C_{s},C_{i},C_{2},C_{2h},C_{2v},C_{3},C_{3h},C_{3v},C_{4},C_{4h},C_{4v},C_{6},C_{6h},C_{6v},D_{2},D_{2d},D_{2h},D_{3},D_{3d},D_{3h},D_{4},D_{4h},D_{6},D_{6h},S_{4},S_{6},T,T_{d},T_{h},O,O_{h}.
T | O | I |
T_{h} | O_{h} | I_{h} |
T_{d} |
This Character Table for the O point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.