D22h E 2 C22 2 C11 2 C22^3 2 C11^2 2 C22^5 2 C11^3 2 C22^7 2 C11^4 2 C22^9 2 C11^5 C2 11 C2' 11 C2" i 2 S22 2 S11 2 S22^3 2 S11^2 2 S22^5 2 S11^3 2 S22^7 2 S11^4 2 S22^9 2 S11^5 sh 11 sv 11 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1g 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ... ... ....T ....... ........T ........... ............T A2g 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 ..T ... ..... ....... ......... ........... ............. B1g 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 -1.0000 -1.0000 1.0000 ... ... ..... ....... ......... ........... ............. B2g 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 -1.0000 1.0000 1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 -1.0000 1.0000 -1.0000 ... ... ..... ....... ......... ........... ............. E1g 2.0000 1.9189 1.6825 1.3097 0.8308 0.2846 -0.2846 -0.8308 -1.3097 -1.6825 -1.9189 -2.0000 0.0000 0.0000 2.0000 -1.9189 -1.6825 -1.3097 -0.8308 -0.2846 0.2846 0.8308 1.3097 1.6825 1.9189 -2.0000 0.0000 0.0000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g 2.0000 1.6825 0.8308 -0.2846 -1.3097 -1.9189 -1.9189 -1.3097 -0.2846 0.8308 1.6825 2.0000 0.0000 0.0000 2.0000 1.6825 0.8308 -0.2846 -1.3097 -1.9189 -1.9189 -1.3097 -0.2846 0.8308 1.6825 2.0000 0.0000 0.0000 ... ... TT... ....... ....TT... ........... ........TT... E3g 2.0000 1.3097 -0.2846 -1.6825 -1.9189 -0.8308 0.8308 1.9189 1.6825 0.2846 -1.3097 -2.0000 0.0000 0.0000 2.0000 -1.3097 0.2846 1.6825 1.9189 0.8308 -0.8308 -1.9189 -1.6825 -0.2846 1.3097 -2.0000 0.0000 0.0000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g 2.0000 0.8308 -1.3097 -1.9189 -0.2846 1.6825 1.6825 -0.2846 -1.9189 -1.3097 0.8308 2.0000 0.0000 0.0000 2.0000 0.8308 -1.3097 -1.9189 -0.2846 1.6825 1.6825 -0.2846 -1.9189 -1.3097 0.8308 2.0000 0.0000 0.0000 ... ... ..... ....... TT....... ........... ....TT....... E5g 2.0000 0.2846 -1.9189 -0.8308 1.6825 1.3097 -1.3097 -1.6825 0.8308 1.9189 -0.2846 -2.0000 0.0000 0.0000 2.0000 -0.2846 1.9189 0.8308 -1.6825 -1.3097 1.3097 1.6825 -0.8308 -1.9189 0.2846 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ..TT......... E6g 2.0000 -0.2846 -1.9189 0.8308 1.6825 -1.3097 -1.3097 1.6825 0.8308 -1.9189 -0.2846 2.0000 0.0000 0.0000 2.0000 -0.2846 -1.9189 0.8308 1.6825 -1.3097 -1.3097 1.6825 0.8308 -1.9189 -0.2846 2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... TT........... E7g 2.0000 -0.8308 -1.3097 1.9189 -0.2846 -1.6825 1.6825 0.2846 -1.9189 1.3097 0.8308 -2.0000 0.0000 0.0000 2.0000 0.8308 1.3097 -1.9189 0.2846 1.6825 -1.6825 -0.2846 1.9189 -1.3097 -0.8308 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E8g 2.0000 -1.3097 -0.2846 1.6825 -1.9189 0.8308 0.8308 -1.9189 1.6825 -0.2846 -1.3097 2.0000 0.0000 0.0000 2.0000 -1.3097 -0.2846 1.6825 -1.9189 0.8308 0.8308 -1.9189 1.6825 -0.2846 -1.3097 2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E9g 2.0000 -1.6825 0.8308 0.2846 -1.3097 1.9189 -1.9189 1.3097 -0.2846 -0.8308 1.6825 -2.0000 0.0000 0.0000 2.0000 1.6825 -0.8308 -0.2846 1.3097 -1.9189 1.9189 -1.3097 0.2846 0.8308 -1.6825 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E10g 2.0000 -1.9189 1.6825 -1.3097 0.8308 -0.2846 -0.2846 0.8308 -1.3097 1.6825 -1.9189 2.0000 0.0000 0.0000 2.0000 -1.9189 1.6825 -1.3097 0.8308 -0.2846 -0.2846 0.8308 -1.3097 1.6825 -1.9189 2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. A1u 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 ... ... ..... ....... ......... ........... ............. A2u 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 1.0000 1.0000 ... ..T ..... ......T ......... ..........T ............. B1u 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 -1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 1.0000 1.0000 -1.0000 ... ... ..... ....... ......... ........... ............. B2u 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 -1.0000 1.0000 -1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 1.0000 -1.0000 1.0000 ... ... ..... ....... ......... ........... ............. E1u 2.0000 1.9189 1.6825 1.3097 0.8308 0.2846 -0.2846 -0.8308 -1.3097 -1.6825 -1.9189 -2.0000 0.0000 0.0000 -2.0000 1.9189 1.6825 1.3097 0.8308 0.2846 -0.2846 -0.8308 -1.3097 -1.6825 -1.9189 2.0000 0.0000 0.0000 ... TT. ..... ....TT. ......... ........TT. ............. E2u 2.0000 1.6825 0.8308 -0.2846 -1.3097 -1.9189 -1.9189 -1.3097 -0.2846 0.8308 1.6825 2.0000 0.0000 0.0000 -2.0000 -1.6825 -0.8308 0.2846 1.3097 1.9189 1.9189 1.3097 0.2846 -0.8308 -1.6825 -2.0000 0.0000 0.0000 ... ... ..... ..TT... ......... ......TT... ............. E3u 2.0000 1.3097 -0.2846 -1.6825 -1.9189 -0.8308 0.8308 1.9189 1.6825 0.2846 -1.3097 -2.0000 0.0000 0.0000 -2.0000 1.3097 -0.2846 -1.6825 -1.9189 -0.8308 0.8308 1.9189 1.6825 0.2846 -1.3097 2.0000 0.0000 0.0000 ... ... ..... TT..... ......... ....TT..... ............. E4u 2.0000 0.8308 -1.3097 -1.9189 -0.2846 1.6825 1.6825 -0.2846 -1.9189 -1.3097 0.8308 2.0000 0.0000 0.0000 -2.0000 -0.8308 1.3097 1.9189 0.2846 -1.6825 -1.6825 0.2846 1.9189 1.3097 -0.8308 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ..TT....... ............. E5u 2.0000 0.2846 -1.9189 -0.8308 1.6825 1.3097 -1.3097 -1.6825 0.8308 1.9189 -0.2846 -2.0000 0.0000 0.0000 -2.0000 0.2846 -1.9189 -0.8308 1.6825 1.3097 -1.3097 -1.6825 0.8308 1.9189 -0.2846 2.0000 0.0000 0.0000 ... ... ..... ....... ......... TT......... ............. E6u 2.0000 -0.2846 -1.9189 0.8308 1.6825 -1.3097 -1.3097 1.6825 0.8308 -1.9189 -0.2846 2.0000 0.0000 0.0000 -2.0000 0.2846 1.9189 -0.8308 -1.6825 1.3097 1.3097 -1.6825 -0.8308 1.9189 0.2846 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E7u 2.0000 -0.8308 -1.3097 1.9189 -0.2846 -1.6825 1.6825 0.2846 -1.9189 1.3097 0.8308 -2.0000 0.0000 0.0000 -2.0000 -0.8308 -1.3097 1.9189 -0.2846 -1.6825 1.6825 0.2846 -1.9189 1.3097 0.8308 2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E8u 2.0000 -1.3097 -0.2846 1.6825 -1.9189 0.8308 0.8308 -1.9189 1.6825 -0.2846 -1.3097 2.0000 0.0000 0.0000 -2.0000 1.3097 0.2846 -1.6825 1.9189 -0.8308 -0.8308 1.9189 -1.6825 0.2846 1.3097 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E9u 2.0000 -1.6825 0.8308 0.2846 -1.3097 1.9189 -1.9189 1.3097 -0.2846 -0.8308 1.6825 -2.0000 0.0000 0.0000 -2.0000 -1.6825 0.8308 0.2846 -1.3097 1.9189 -1.9189 1.3097 -0.2846 -0.8308 1.6825 2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E10u 2.0000 -1.9189 1.6825 -1.3097 0.8308 -0.2846 -0.2846 0.8308 -1.3097 1.6825 -1.9189 2.0000 0.0000 0.0000 -2.0000 1.9189 -1.6825 1.3097 -0.8308 0.2846 0.2846 -0.8308 1.3097 -1.6825 1.9189 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.918985947229 = 2*cos(2*π/22) = 2*cos(π/11) 1.682507065662 = 2*cos(4*π/22) = 2*cos(2*π/11) 1.309721467891 = 2*cos(6*π/22) = 2*cos(3*π/11) 0.830830026004 = 2*cos(8*π/22) = 2*cos(4*π/11) 0.284629676547 = 2*cos(10*π/22) = 2*cos(5*π/11) Symmetry of Rotations and Cartesian products A1g d+g+i+k+m z2, z4, z6 A2g R Rz E1g R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} E2g d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E3g g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4g g+i+k+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5g i+k+m {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6g i+k+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7g k+m E8g k+m E9g m E10g m A2u p+f+h+j+l z, z3, z5 E1u p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E2u f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E3u f+h+j+l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E4u h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E5u h+j+l {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)} E6u j+l E7u j+l E8u l E9u l Notes: α The order of the D22h point group is 88, and the order of the principal axis (C22) is 22. The group has 28 irreducible representations. β The D22h point group is generated by three symmetry elements that are canonically chosen C22, 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 S22 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 D22h 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 Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. For this group, however, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.
| D20h | ||
| D21h | ||
| C22 C22v C22h D22 | D22h | D22d S22 |
| D23h | ||
| D24h |
This Character Table for the D22h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.