D22d E 2 S44 2 C22 2 S44^3 2 C11 2 S44^5 2 C22^3 2 S44^7 2 C11^2 2 S44^9 2 C22^5 2 S4 2 C11^3 2 S44^13 2 C22^7 2 S44^15 2 C11^4 2 S44^17 2 C22^9 2 S44^19 2 C11^5 2 S44^21 C2 22 C2' 22 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ... ... ....T ....... ........T ........... ............T A2 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 ..T ... ..... ....... ......... ........... ............. B1 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. B2 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 ... ..T ..... ......T ......... ..........T ............. E1 2.00000 1.97964 1.91899 1.81926 1.68251 1.51150 1.30972 1.08128 0.83083 0.56347 0.28463 0.00000 -0.28463 -0.56347 -0.83083 -1.08128 -1.30972 -1.51150 -1.68251 -1.81926 -1.91899 -1.97964 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 2.00000 1.91899 1.68251 1.30972 0.83083 0.28463 -0.28463 -0.83083 -1.30972 -1.68251 -1.91899 -2.00000 -1.91899 -1.68251 -1.30972 -0.83083 -0.28463 0.28463 0.83083 1.30972 1.68251 1.91899 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 2.00000 1.81926 1.30972 0.56347 -0.28463 -1.08128 -1.68251 -1.97964 -1.91899 -1.51150 -0.83083 0.00000 0.83083 1.51150 1.91899 1.97964 1.68251 1.08128 0.28463 -0.56347 -1.30972 -1.81926 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5 2.00000 1.51150 0.28463 -1.08128 -1.91899 -1.81926 -0.83083 0.56347 1.68251 1.97964 1.30972 0.00000 -1.30972 -1.97964 -1.68251 -0.56347 0.83083 1.81926 1.91899 1.08128 -0.28463 -1.51150 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6 2.00000 1.30972 -0.28463 -1.68251 -1.91899 -0.83083 0.83083 1.91899 1.68251 0.28463 -1.30972 -2.00000 -1.30972 0.28463 1.68251 1.91899 0.83083 -0.83083 -1.91899 -1.68251 -0.28463 1.30972 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.08128 -0.83083 -1.97964 -1.30972 0.56347 1.91899 1.51150 -0.28463 -1.81926 -1.68251 0.00000 1.68251 1.81926 0.28463 -1.51150 -1.91899 -0.56347 1.30972 1.97964 0.83083 -1.08128 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 0.56347 -1.68251 -1.51150 0.83083 1.97964 0.28463 -1.81926 -1.30972 1.08128 1.91899 0.00000 -1.91899 -1.08128 1.30972 1.81926 -0.28463 -1.97964 -0.83083 1.51150 1.68251 -0.56347 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 0.28463 -1.91899 -0.83083 1.68251 1.30972 -1.30972 -1.68251 0.83083 1.91899 -0.28463 -2.00000 -0.28463 1.91899 0.83083 -1.68251 -1.30972 1.30972 1.68251 -0.83083 -1.91899 0.28463 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 -0.56347 -1.68251 1.51150 0.83083 -1.97964 0.28463 1.81926 -1.30972 -1.08128 1.91899 0.00000 -1.91899 1.08128 1.30972 -1.81926 -0.28463 1.97964 -0.83083 -1.51150 1.68251 0.56347 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 -0.83083 -1.30972 1.91899 -0.28463 -1.68251 1.68251 0.28463 -1.91899 1.30972 0.83083 -2.00000 0.83083 1.30972 -1.91899 0.28463 1.68251 -1.68251 -0.28463 1.91899 -1.30972 -0.83083 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 -1.08128 -0.83083 1.97964 -1.30972 -0.56347 1.91899 -1.51150 -0.28463 1.81926 -1.68251 0.00000 1.68251 -1.81926 0.28463 1.51150 -1.91899 0.56347 1.30972 -1.97964 0.83083 1.08128 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 -1.51150 0.28463 1.08128 -1.91899 1.81926 -0.83083 -0.56347 1.68251 -1.97964 1.30972 0.00000 -1.30972 1.97964 -1.68251 0.56347 0.83083 -1.81926 1.91899 -1.08128 -0.28463 1.51150 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E18 2.00000 -1.68251 0.83083 0.28463 -1.30972 1.91899 -1.91899 1.30972 -0.28463 -0.83083 1.68251 -2.00000 1.68251 -0.83083 -0.28463 1.30972 -1.91899 1.91899 -1.30972 0.28463 0.83083 -1.68251 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ..TT....... ............. E19 2.00000 -1.81926 1.30972 -0.56347 -0.28463 1.08128 -1.68251 1.97964 -1.91899 1.51150 -0.83083 0.00000 0.83083 -1.51150 1.91899 -1.97964 1.68251 -1.08128 0.28463 0.56347 -1.30972 1.81926 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E20 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E21 2.00000 -1.97964 1.91899 -1.81926 1.68251 -1.51150 1.30972 -1.08128 0.83083 -0.56347 0.28463 0.00000 -0.28463 0.56347 -0.83083 1.08128 -1.30972 1.51150 -1.68251 1.81926 -1.91899 1.97964 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.979642883762 = 2*cos(2*π/44) = 2*cos(π/22) 1.918985947229 = 2*cos(4*π/44) = 2*cos(π/11) 1.819263990709 = 2*cos(6*π/44) = 2*cos(3*π/22) 1.682507065662 = 2*cos(8*π/44) = 2*cos(2*π/11) 1.511499148709 = 2*cos(10*π/44) = 2*cos(5*π/22) 1.309721467891 = 2*cos(12*π/44) = 2*cos(3*π/11) 1.081281634911 = 2*cos(14*π/44) = 2*cos(7*π/22) 0.830830026004 = 2*cos(16*π/44) = 2*cos(4*π/11) 0.563465113683 = 2*cos(18*π/44) = 2*cos(9*π/22) 0.284629676547 = 2*cos(20*π/44) = 2*cos(5*π/11) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z2, z4, z6 A2 R Rz B2 p+f+h+j+l z, z3, z5 E1 p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E2 d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E3 f+h+j+l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E4 g+i+k+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 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)} E6 i+k+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+l E8 k+m E9 l E10 m E13 m E14 l E15 k+m E16 j+l E17 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)} E18 h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E19 g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E20 f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E21 R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} Notes: α The order of the D22d point group is 88, and the order of the principal axis (S44) is 44. The group has 25 irreducible representations. β The D22d point group is isomorphic to C44v and D44. γ The D22d point group is generated by two symmetry elements, S44 and either a perpendicular C2′ or a vertical σd. Also, the group may be generated from a C2′ plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The group contains one set of twofold symmetry axes (C2′) perpendicular to the principal (z) axis. Both x and y axes are members of that set. ε The single σd set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z. ζ The lowest nonvanishing multipole moment in D22d is 4 (quadrupole moment). η This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. 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.
| D20d | ||
| D21d | ||
| C22 C22v C22h D22 D22h | D22d | S22 |
| D23d | ||
| D24d |
This Character Table for the D22d point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.