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 z2, (x2−y2)2−4x2y2, z4, z2((x2−y2)2−4x2y2), z6 A2g R+g+i+2k+2m Rz, xy(x2−y2), xyz2(x2−y2) B1g d+g+2i+2k+3m x2−y2, z2(x2−y2), x2(x2−3y2)2−y2(3x2−y2)2, z4(x2−y2) B2g d+g+2i+2k+3m xy, xyz2, xy(x2−3y2)(3x2−y2), xyz4 Eg R+d+2g+3i+4k+5m {Rx, Ry}, {xz, yz}, {xz(x2−3y2), yz(3x2−y2)}, {xz3, yz3}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)}, {xz5, yz5} A1u h+j+2l xyz(x2−y2) A2u p+f+2h+2j+3l z, z3, z((x2−y2)2−4x2y2), z5 B1u f+h+2j+2l xyz, xyz3 B2u f+h+2j+2l z(x2−y2), z3(x2−y2) Eu p+2f+3h+4j+5l {x, y}, {x(x2−3y2), y(3x2−y2)}, {xz2, yz2}, {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {xz4, yz4} Notes: α The order of the D4h point group is 16, and the order of the principal axis (C4) is 4. The group has 10 irreducible representations. β The D4h point group is generated by three symmetry elements that are canonically chosen C4, C2′ and i. Other choices include σh instead of i, or any of C2″, σv or σd instead of C2′. Also, some ternary combinations of C2′, C2″, σv and σd act as generators. Lastly, the S4 can be chosen, together with i or σh and any one of C2′, C2″, σv or σd. 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 C2′ 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 σv contains both the xz and the yz planes. ε The lowest nonvanishing multipole moment in D4h 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: C1,Cs,Ci,C2,C2h,C2v,C3,C3h,C3v,C4,C4h,C4v,C6,C6h,C6v,D2,D2d,D2h,D3,D3d,D3h,D4,D4h,D6,D6h,S4,S6,T,Td,Th,O,Oh.
| D2h | ||
| D3h | ||
| C4 C4v C4h D4 | D4h | D4d S4 |
| D5h | ||
| D6h |
This Character Table for the D4h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.