Th E 8 C3 3 C2 i 8 S6 3 sh <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> Ag 1 1 1 1 1 1 ... ... ..... ....... T........ ........... TT........... Eg * 2 -1 2 2 -1 2 ... ... TT... ....... .TT...... ........... ..TT......... Tg 3 0 -1 3 0 -1 TTT ... ..TTT ....... ...TTTTTT ........... ....TTTTTTTTT Au 1 1 1 -1 -1 -1 ... ... ..... T...... ......... ........... ............. Eu * 2 -1 2 -2 1 -2 ... ... ..... ....... ......... TT......... ............. Tu 3 0 -1 -3 0 1 ... TTT ..... .TTTTTT ......... ..TTTTTTTTT ............. Symmetry of Rotations and Cartesian products Ag g+2i+k+2m x^{4}+y^{4}+z^{4}, x^{2}y^{2}z^{2}, x^{4}(y^{2}−z^{2})+y^{4}(z^{2}−x^{2})+z^{4}(x^{2}−y^{2}) Eg d+g+i+2k+2m {x^{2}−y^{2}, 2z^{2}−x^{2}−y^{2}}, {x^{4}−y^{4}, 2z^{4}−x^{4}−y^{4}}, {2z^{6}−x^{6}−y^{6}, x^{6}−y^{6}} Tg R+d+2g+3i+4k+5m {R_{x}, R_{y}, R_{z}}, {xy, xz, yz}, {xy(x^{2}−y^{2}), xz(x^{2}−z^{2}), yz(y^{2}−z^{2})}, {x^{2}yz, xy^{2}z, xyz^{2}}, {xy(x^{4}−y^{4}), xz(x^{4}−z^{4}), yz(y^{4}−z^{4})}, {x^{4}yz, xy^{4}z, xyz^{4}}, {x^{3}y^{3}, x^{3}z^{3}, y^{3}z^{3}} Au f+j+2l xyz Eu h+j+l {xyz(x^{2}−y^{2}), xyz(2z^{2}−x^{2}−y^{2})} Tu p+2f+3h+4j+5l {x, y, z}, {x(z^{2}−y^{2}), y(z^{2}−x^{2}), z(x^{2}−y^{2})}, {x^{3}, y^{3}, z^{3}}, {x^{2}y^{2}z, x^{2}yz^{2}, xy^{2}z^{2}}, {x^{5}, y^{5}, z^{5}}, {x(x^{4}−z^{4}), y(x^{4}−z^{4}), z(x^{4}−y^{4})} Notes: α The order of the T_{h} point group is 24, and the order of the principal axis (S_{6}) is 6. The group has 6 irreducible representations. β The T_{h} point group is generated by two symmetry elements, which can be chosen as two distinct S_{6} axes, or an S_{6} with either C_{2} or σ_{h}, or C_{3} and σ_{h}. γ The lowest nonvanishing multipole moment in T_{h} is 16 (hexadecapole moment). δ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. ε The point group contains four complex-valued one-dimensional irreducible representations that have been combined into E_{g} and E_{u}. These “E representations” are reducible, but have the advantage of real (and integer) character values. Also, the “8 C_{3}” and “8 S_{6}” pseudo-classes are the merger of four true classes containing separate left and right rotations. ζ The 2 reducible “E” representations almost behave like true irreducible representations. Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table. This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula. 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 tetrahedral because it has no four-fold axis 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 T_{h} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.