C22h 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 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 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> Ag 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 ........... ............T Bg 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 2.0000 -1.9189 -1.6825 -1.3097 -0.8308 -0.2846 0.2846 0.8308 1.3097 1.6825 1.9189 -2.0000 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 2.0000 1.6825 0.8308 -0.2846 -1.3097 -1.9189 -1.9189 -1.3097 -0.2846 0.8308 1.6825 2.0000 ... ... 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 2.0000 -1.3097 0.2846 1.6825 1.9189 0.8308 -0.8308 -1.9189 -1.6825 -0.2846 1.3097 -2.0000 ... ... ..... ....... ..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 2.0000 0.8308 -1.3097 -1.9189 -0.2846 1.6825 1.6825 -0.2846 -1.9189 -1.3097 0.8308 2.0000 ... ... ..... ....... 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 2.0000 -0.2846 1.9189 0.8308 -1.6825 -1.3097 1.3097 1.6825 -0.8308 -1.9189 0.2846 -2.0000 ... ... ..... ....... ......... ........... ..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 2.0000 -0.2846 -1.9189 0.8308 1.6825 -1.3097 -1.3097 1.6825 0.8308 -1.9189 -0.2846 2.0000 ... ... ..... ....... ......... ........... 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 2.0000 0.8308 1.3097 -1.9189 0.2846 1.6825 -1.6825 -0.2846 1.9189 -1.3097 -0.8308 -2.0000 ... ... ..... ....... ......... ........... ............. 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 2.0000 -1.3097 -0.2846 1.6825 -1.9189 0.8308 0.8308 -1.9189 1.6825 -0.2846 -1.3097 2.0000 ... ... ..... ....... ......... ........... ............. 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 2.0000 1.6825 -0.8308 -0.2846 1.3097 -1.9189 1.9189 -1.3097 0.2846 0.8308 -1.6825 -2.0000 ... ... ..... ....... ......... ........... ............. 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 2.0000 -1.9189 1.6825 -1.3097 0.8308 -0.2846 -0.2846 0.8308 -1.3097 1.6825 -1.9189 2.0000 ... ... ..... ....... ......... ........... ............. Au 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 ............. Bu 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 -2.0000 1.9189 1.6825 1.3097 0.8308 0.2846 -0.2846 -0.8308 -1.3097 -1.6825 -1.9189 2.0000 ... 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 -2.0000 -1.6825 -0.8308 0.2846 1.3097 1.9189 1.9189 1.3097 0.2846 -0.8308 -1.6825 -2.0000 ... ... ..... ..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 -2.0000 1.3097 -0.2846 -1.6825 -1.9189 -0.8308 0.8308 1.9189 1.6825 0.2846 -1.3097 2.0000 ... ... ..... 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 -2.0000 -0.8308 1.3097 1.9189 0.2846 -1.6825 -1.6825 0.2846 1.9189 1.3097 -0.8308 -2.0000 ... ... ..... ....... ......... ..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 -2.0000 0.2846 -1.9189 -0.8308 1.6825 1.3097 -1.3097 -1.6825 0.8308 1.9189 -0.2846 2.0000 ... ... ..... ....... ......... 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 -2.0000 0.2846 1.9189 -0.8308 -1.6825 1.3097 1.3097 -1.6825 -0.8308 1.9189 0.2846 -2.0000 ... ... ..... ....... ......... ........... ............. 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 -2.0000 -0.8308 -1.3097 1.9189 -0.2846 -1.6825 1.6825 0.2846 -1.9189 1.3097 0.8308 2.0000 ... ... ..... ....... ......... ........... ............. 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 -2.0000 1.3097 0.2846 -1.6825 1.9189 -0.8308 -0.8308 1.9189 -1.6825 0.2846 1.3097 -2.0000 ... ... ..... ....... ......... ........... ............. 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 -2.0000 -1.6825 0.8308 0.2846 -1.3097 1.9189 -1.9189 1.3097 -0.2846 -0.8308 1.6825 2.0000 ... ... ..... ....... ......... ........... ............. 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 -2.0000 1.9189 -1.6825 1.3097 -0.8308 0.2846 0.2846 -0.8308 1.3097 -1.6825 1.9189 -2.0000 ... ... ..... ....... ......... ........... ............. 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 Ag R+d+g+i+k+m Rz, z2, z4, z6 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 Au 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 C22h point group is 44, and the order of the principal axis (C22) is 22. The group has 24 irreducible representations. β The C22h point group is generated by two symmetry elements, which are canonically chosen as C22 and i. Other possible choices are C22 and σh, or less commonly S22 with either C2 or σh. γ The lowest nonvanishing multipole moment in C22h is 4 (quadrupole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The C22h group is Abelian because all its symmetry operations are coaxial. This is a sufficient condition. In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. ε Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 40 cases have been combined into 20 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 20 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ζ The 20 reducible “E” representations almost behave like true irreducible representations. Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table. This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula. 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.
| C20h | ||
| C21h | ||
| C22 C22v | C22h | D22 D22h D22d S22 |
| C23h | ||
| C24h |
This Character Table for the C22h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.