Character table for the C22v point group

C22v    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 sv   11 sd      <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 ........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 ... ..... ....... ......... ........... .............
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   p+d+f+g+h+i+j+k+l+m    z, z2, z3, z4, z5, z6 
A2   R                      Rz 
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      {x2y2, xy}, {z(x2y2), xyz}, {z2(x2y2), xyz2}, {z3(x2y2), xyz3}, {z4(x2y2), xyz4} 
E3   f+g+h+i+j+k+l+m        {x(x2−3y2), y(3x2y2)}, {xz(x2−3y2), yz(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4   g+h+i+j+k+l+m          {(x2y2)2−4x2y2, xy(x2y2)}, {z((x2y2)2−4x2y2), xyz(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E5   h+i+j+k+l+m            {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6   i+j+k+l+m              {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E7   j+k+l+m 
E8   k+l+m 
E9   l+m 
E10  m 

 Notes:

    α  The order of the C22v point group is 44, and the order of the principal axis (C22) is 22. The group has 14 irreducible representations.

    β  The C22v point group is isomorphic to D11d, D11h and D22.

    γ  The C22v point group is generated by two symmetry elements, C22 and any σv (or, non-canonically, any σd).
       Also, the group may be generated from a σv plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe).

    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 C22v is 2 (dipole 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.

This Character Table for the C22v point group was created by Gernot Katzer.

For other groups and some explanations, see the Main Page.