D22 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" <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 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 A2 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 ......... ..........T ............. B1 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 ... ... ..... ....... ......... ........... ............. B2 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 ... ... ..... ....... ......... ........... ............. E1 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. ......TT. ........TT. ..........TT. E2 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... ......TT... ........TT... E3 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..... ....TT..... ......TT..... E4 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....... ....TT....... E5 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......... ..TT......... E6 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........... E7 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 ... ... ..... ....... ......... ........... ............. E8 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 ... ... ..... ....... ......... ........... ............. E9 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 ... ... ..... ....... ......... ........... ............. E10 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 A1 d+g+i+k+m z2, z4, z6 A2 R+p+f+h+j+l Rz, z, z3, z5 E1 R+p+d+f+g+h+i+j+k+l+m {Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {xz3, yz3}, {xz4, yz4}, {xz5, yz5} E2 d+f+g+h+i+j+k+l+m {x2−y2, xy}, {z(x2−y2), xyz}, {z2(x2−y2), xyz2}, {z3(x2−y2), xyz3}, {z4(x2−y2), xyz4} E3 f+g+h+i+j+k+l+m {x(x2−3y2), y(3x2−y2)}, {xz(x2−3y2), yz(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4 g+h+i+j+k+l+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 h+i+j+k+l+m {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6 i+j+k+l+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the D22 point group is 44, and the order of the principal axis (C22) is 22. The group has 14 irreducible representations. β The D22 point group is isomorphic to D11d, D11h and C22v. γ The D22 point group is generated by two symmetry elements, C22 and a perpendicular C2′ (or, non-canonically, C2″). Also, the group may be generated from a C2′ plus a C2″ (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ 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. ε The lowest nonvanishing multipole moment in D22 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 The point group is chiral, as it does not contain any mirroring operation. θ 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.
| D20 | ||
| D21 | ||
| C22 C22v C22h | D22 | D22h D22d S22 |
| D23 | ||
| D24 |
This Character Table for the D22 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.