T E 8 C3 3 C2 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A 1 1 1 ... ... ..... T...... T........ ........... TT........... E * 2 -1 2 ... ... TT... ....... .TT...... TT......... ..TT......... T 3 0 -1 TTT TTT ..TTT .TTTTTT ...TTTTTT ..TTTTTTTTT ....TTTTTTTTT Symmetry of Rotations and Cartesian products A f+g+2i+j+k+2l+2m xyz, 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}) 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}} T R+p+d+2f+2g+3h+3i+4j+4k+5l+5m {R_{x}, R_{y}, R_{z}}, {x, y, z}, {xy, xz, yz}, {x(z^{2}−y^{2}), y(z^{2}−x^{2}), z(x^{2}−y^{2})}, {x^{3}, y^{3}, z^{3}}, {xy(x^{2}−y^{2}), xz(x^{2}−z^{2}), yz(y^{2}−z^{2})}, {x^{2}yz, xy^{2}z, xyz^{2}}, {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})}, {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}} Notes: α The order of the T point group is 12, and the order of the principal axis (C_{3}) is 3. The group has 3 irreducible representations. β The T point group is isomorphic to the Alternating Group Alt(4). It is not a simple group, as it contains the Klein four-group D_{2} as a normal subgroup (with quotient group C_{3}). γ The T point group is generated by two symmetry elements, which can be chosen as two distinct C_{3} axes, or any C_{3} with any C_{2}. δ The lowest nonvanishing multipole moment in T is 8 (octupole 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 contains two complex-valued one-dimensional irreducible representations that have been combined and labeled E. This “E representation” is reducible, but has the advantage of real (and integer) character values. Also, the “8 C_{3}” pseudo-class is the merger of two true classes, one containing four right and the other containing four left rotations. η The single “E” representation is reducible but almost behaves like a true irreducible representation. Its norm, however, is twice the group order. Therefore, is has 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. θ The point group is chiral, as it does not contain any mirroring operation. ι 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. κ 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. λ 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 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.