D3h E 2 C3 3 C2' sh 2 S3 3 sv <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1' 1 1 1 1 1 1 ... ... ....T T...... ........T ....T...... T...........T A1" 1 1 1 -1 -1 -1 ... ... ..... ....... ...T..... ........... .......T..... A2' 1 1 -1 1 1 -1 ..T ... ..... .T..... ......... .....T..... .T........... A2" 1 1 -1 -1 -1 1 ... ..T ..... ......T ..T...... ..........T ......T...... E' 2 -1 0 2 -1 0 ... TT. TT... ....TT. TT..TT... TT......TT. ....TT..TT... E" 2 -1 0 -2 1 0 TT. ... ..TT. ..TT... ......TT. ..TT..TT... ..TT......TT. Symmetry of Rotations and Cartesian products A1' d+f+g+h+2i+j+2k+2l+2m z^{2}, x(x^{2}−3y^{2}), z^{4}, xz^{2}(x^{2}−3y^{2}), x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, z^{6} A1" g+i+j+k+l+2m yz(3x^{2}−y^{2}), yz^{3}(3x^{2}−y^{2}) A2' R+f+h+i+j+k+2l+m R_{z}, y(3x^{2}−y^{2}), yz^{2}(3x^{2}−y^{2}), xy(x^{2}−3y^{2})(3x^{2}−y^{2}) A2" p+f+g+h+i+2j+k+2l+2m z, z^{3}, xz(x^{2}−3y^{2}), z^{5}, xz^{3}(x^{2}−3y^{2}) E' p+d+f+2g+2h+2i+3j+3k+3l+4m {x, y}, {x^{2}−y^{2}, xy}, {xz^{2}, yz^{2}}, {(x^{2}−y^{2})^{2}−4x^{2}y^{2}, xy(x^{2}−y^{2})}, {z^{2}(x^{2}−y^{2}), xyz^{2}}, {x(x^{2}−(5+2√5)y^{2})(x^{2}−(5−2√5)y^{2}), y((5+2√5)x^{2}−y^{2})((5−2√5)x^{2}−y^{2})}, {xz^{4}, yz^{4}}, {z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz^{2}(x^{2}−y^{2})}, {z^{4}(x^{2}−y^{2}), xyz^{4}} E" R+d+f+g+2h+2i+2j+3k+3l+3m {R_{x}, R_{y}}, {xz, yz}, {z(x^{2}−y^{2}), xyz}, {xz^{3}, yz^{3}}, {z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz(x^{2}−y^{2})}, {z^{3}(x^{2}−y^{2}), xyz^{3}}, {xz(x^{2}−(5+2√5)y^{2})(x^{2}−(5−2√5)y^{2}), yz((5+2√5)x^{2}−y^{2})((5−2√5)x^{2}−y^{2})}, {xz^{5}, yz^{5}} Notes: α The order of the D_{3h} point group is 12, and the order of the principal axis (S_{3}) is 6. The group has 6 irreducible representations. β The D_{3h} point group is isomorphic to D_{3d}, C_{6v} and D_{6}. γ The D_{3h} point group is generated by two symmetry elements, S_{3} and either a perpendicular C_{2}^{′} or a vertical σ_{v}. Also, the group may be generated from any two σ_{v} planes, or any σ_{v} and a non-coplanar C_{2}^{′}. The canonical choice, however, is to use redundant generators: C_{3}, C_{2}^{′} and σ_{h}. δ The group contains one set of C_{2}^{′} symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set. Similarly, the single set of symmetry planes denoted σ_{d} contains the xz plane but not the yz plane. ε The lowest nonvanishing multipole moment in D_{3h} is 4 (quadrupole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η All characters are integers because the order of the principal axis is 1,2,3,4 or 6. This implies that the point group corresponds to a constructible polygon which can be used for tiling the plane. 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}.
D_{1h} | ||
D_{2h} | ||
C_{3} C_{3v} C_{3h} D_{3} | D_{3h} | D_{3d} |
D_{4h} | ||
D_{5h} |
This Character Table for the D_{3h} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.