D6h E 2 C6 2 C3 C2 3 C2' 3 C2" i 2 S6 2 S3 sh 3 sv 3 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1g 1 1 1 1 1 1 1 1 1 1 1 1 ... ... ....T ....... ........T ........... T...........T A2g 1 1 1 1 -1 -1 1 1 1 1 -1 -1 ..T ... ..... ....... ......... ........... .T........... B1g 1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 ... ... ..... ....... ...T..... ........... .......T..... B2g 1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 ... ... ..... ....... ..T...... ........... ......T...... E1g 2 1 -1 -2 0 0 2 -1 1 -2 0 0 TT. ... ..TT. ....... ......TT. ........... ..TT......TT. E2g 2 -1 -1 2 0 0 2 -1 -1 2 0 0 ... ... TT... ....... TT..TT... ........... ....TT..TT... A1u 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 ... ... ..... ....... ......... ........... ............. A2u 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 ... ..T ..... ......T ......... ..........T ............. B1u 1 -1 1 -1 1 -1 -1 -1 1 1 1 -1 ... ... ..... T...... ......... ....T...... ............. B2u 1 -1 1 -1 -1 1 -1 -1 1 1 -1 1 ... ... ..... .T..... ......... .....T..... ............. E1u 2 1 -1 -2 0 0 -2 1 -1 2 0 0 ... TT. ..... ....TT. ......... TT......TT. ............. E2u 2 -1 -1 2 0 0 -2 1 1 -2 0 0 ... ... ..... ..TT... ......... ..TT..TT... ............. Symmetry of Rotations and Cartesian products A1g d+g+2i+2k+2m z2, z4, x2(x2−3y2)2−y2(3x2−y2)2, z6 A2g R+i+k+m Rz, xy(x2−3y2)(3x2−y2) B1g g+i+k+2m yz(3x2−y2), yz3(3x2−y2) B2g g+i+k+2m xz(x2−3y2), xz3(x2−3y2) E1g R+d+g+2i+3k+3m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz5, yz5} E2g d+2g+2i+3k+4m {x2−y2, xy}, {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2(x2−y2), xyz2}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)}, {z4(x2−y2), xyz4} A1u j+l A2u p+f+h+2j+2l z, z3, z5 B1u f+h+j+2l x(x2−3y2), xz2(x2−3y2) B2u f+h+j+2l y(3x2−y2), yz2(3x2−y2) E1u p+f+2h+3j+3l {x, y}, {xz2, yz2}, {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz4, yz4} E2u f+2h+2j+3l {z(x2−y2), xyz}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {z3(x2−y2), xyz3} Notes: α The order of the D6h point group is 24, and the order of the principal axis (C6) is 6. The group has 12 irreducible representations. β The D6h point group is generated by three symmetry elements that are canonically chosen C6, 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 S6 can be chosen, together with σh or C2 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′ has the x axis as a member, while the y axis is a member of the C2″ set. 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 has the xz plane as a member, while the yz plane is a member of the σd set. ε The lowest nonvanishing multipole moment in D6h 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.
| D4h | ||
| D5h | ||
| C6 C6v C6h D6 | D6h | D6d S6 |
| D7h | ||
| D8h |
This Character Table for the D6h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.