D4h E 2 C4 C2 2 C2' 2 C2" i 2 S4 sh 2 sv 2 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1g 1 1 1 1 1 1 1 1 1 1 ... ... ....T ....... T.......T ........... ....T.......T A2g 1 1 1 -1 -1 1 1 1 -1 -1 ..T ... ..... ....... .T....... ........... .....T....... B1g 1 -1 1 1 -1 1 -1 1 1 -1 ... ... T.... ....... ....T.... ........... T.......T.... B2g 1 -1 1 -1 1 1 -1 1 -1 1 ... ... .T... ....... .....T... ........... .T.......T... Eg 2 0 -2 0 0 2 0 -2 0 0 TT. ... ..TT. ....... ..TT..TT. ........... ..TT..TT..TT. A1u 1 1 1 1 1 -1 -1 -1 -1 -1 ... ... ..... ....... ......... ...T....... ............. A2u 1 1 1 -1 -1 -1 -1 -1 1 1 ... ..T ..... ......T ......... ..T.......T ............. B1u 1 -1 1 1 -1 -1 1 -1 -1 1 ... ... ..... ...T... ......... .......T... ............. B2u 1 -1 1 -1 1 -1 1 -1 1 -1 ... ... ..... ..T.... ......... ......T.... ............. Eu 2 0 -2 0 0 -2 0 2 0 0 ... TT. ..... TT..TT. ......... TT..TT..TT. ............. Symmetry of Rotations and Cartesian products A1g d+2g+2i+3k+3m z^{2}, (x^{2}−y^{2})^{2}−4x^{2}y^{2}, z^{4}, z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), z^{6} A2g R+g+i+2k+2m R_{z}, xy(x^{2}−y^{2}), xyz^{2}(x^{2}−y^{2}) B1g d+g+2i+2k+3m x^{2}−y^{2}, z^{2}(x^{2}−y^{2}), x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, z^{4}(x^{2}−y^{2}) B2g d+g+2i+2k+3m xy, xyz^{2}, xy(x^{2}−3y^{2})(3x^{2}−y^{2}), xyz^{4} Eg R+d+2g+3i+4k+5m {R_{x}, R_{y}}, {xz, yz}, {xz(x^{2}−3y^{2}), yz(3x^{2}−y^{2})}, {xz^{3}, yz^{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^{3}(x^{2}−3y^{2}), yz^{3}(3x^{2}−y^{2})}, {xz^{5}, yz^{5}} A1u h+j+2l xyz(x^{2}−y^{2}) A2u p+f+2h+2j+3l z, z^{3}, z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), z^{5} B1u f+h+2j+2l xyz, xyz^{3} B2u f+h+2j+2l z(x^{2}−y^{2}), z^{3}(x^{2}−y^{2}) Eu p+2f+3h+4j+5l {x, y}, {x(x^{2}−3y^{2}), y(3x^{2}−y^{2})}, {xz^{2}, yz^{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^{2}(x^{2}−3y^{2}), yz^{2}(3x^{2}−y^{2})}, {xz^{4}, yz^{4}} Notes: α The order of the D_{4h} point group is 16, and the order of the principal axis (C_{4}) is 4. The group has 10 irreducible representations. β The D_{4h} point group is generated by three symmetry elements that are canonically chosen C_{4}, C_{2}^{′} and i. Other choices include σ_{h} instead of i, or any of C_{2}^{″}, σ_{v} or σ_{d} instead of C_{2}^{′}. Also, some ternary combinations of C_{2}^{′}, C_{2}^{″}, σ_{v} and σ_{d} act as generators. Lastly, the S_{4} can be chosen, together with i or σ_{h} and any one of C_{2}^{′}, C_{2}^{″}, σ_{v} or σ_{d}. γ There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation). By convention, the set denoted as C_{2}^{′} contains both the x and y axes. δ There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation). By convention, the set denoted as σ_{v} contains both the xz and the yz planes. ε The lowest nonvanishing multipole moment in D_{4h} 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_{2h} | ||
D_{3h} | ||
C_{4} C_{4v} C_{4h} D_{4} | D_{4h} | D_{4d} S_{4} |
D_{5h} | ||
D_{6h} |
This Character Table for the D_{4h} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.