Character table for the D11d point group

D11d    E       2 C11   2 C11^2 2 C11^3 2 C11^4 2 C11^5 11 C2'  i       2 S22   2 S22^3 2 S22^5 2 S22^7 2 S22^9 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     ... ... ....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     ..T ... ..... ....... ......... ........... .............
E1g     2.0000  1.6825  0.8308 -0.2846 -1.3097 -1.9189  0.0000  2.0000 -1.9189 -1.3097 -0.2846  0.8308  1.6825  0.0000     TT. ... ..TT. ....... ......TT. ........... ..........TT.
E2g     2.0000  0.8308 -1.3097 -1.9189 -0.2846  1.6825  0.0000  2.0000  1.6825 -0.2846 -1.9189 -1.3097  0.8308  0.0000     ... ... TT... ....... ....TT... ........... ........TT...
E3g     2.0000 -0.2846 -1.9189  0.8308  1.6825 -1.3097  0.0000  2.0000 -1.3097  1.6825  0.8308 -1.9189 -0.2846  0.0000     ... ... ..... ....... ..TT..... ........... ......TT.....
E4g     2.0000 -1.3097 -0.2846  1.6825 -1.9189  0.8308  0.0000  2.0000  0.8308 -1.9189  1.6825 -0.2846 -1.3097  0.0000     ... ... ..... ....... TT....... ........... ....TT.......
E5g     2.0000 -1.9189  1.6825 -1.3097  0.8308 -0.2846  0.0000  2.0000 -0.2846  0.8308 -1.3097  1.6825 -1.9189  0.0000     ... ... ..... ....... ......... ........... TTTT.........
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     ... ... ..... ....... ......... ........... .............
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     ... ..T ..... ......T ......... ..........T .............
E1u     2.0000  1.6825  0.8308 -0.2846 -1.3097 -1.9189  0.0000 -2.0000  1.9189  1.3097  0.2846 -0.8308 -1.6825  0.0000     ... TT. ..... ....TT. ......... ........TT. .............
E2u     2.0000  0.8308 -1.3097 -1.9189 -0.2846  1.6825  0.0000 -2.0000 -1.6825  0.2846  1.9189  1.3097 -0.8308  0.0000     ... ... ..... ..TT... ......... ......TT... .............
E3u     2.0000 -0.2846 -1.9189  0.8308  1.6825 -1.3097  0.0000 -2.0000  1.3097 -1.6825 -0.8308  1.9189  0.2846  0.0000     ... ... ..... TT..... ......... ....TT..... .............
E4u     2.0000 -1.3097 -0.2846  1.6825 -1.9189  0.8308  0.0000 -2.0000 -0.8308  1.9189 -1.6825  0.2846  1.3097  0.0000     ... ... ..... ....... ......... ..TT....... .............
E5u     2.0000 -1.9189  1.6825 -1.3097  0.8308 -0.2846  0.0000 -2.0000  0.2846 -0.8308  1.3097 -1.6825  1.9189  0.0000     ... ... ..... ....... ......... TT......... .............

 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+2m  {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} 
E2g  d+g+i+k+2m    {x2y2, xy}, {z2(x2y2), xyz2}, {z4(x2y2), xyz4} 
E3g  g+i+2k+2m     {xz(x2−3y2), yz(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4g  g+i+2k+2m     {(x2y2)2−4x2y2, xy(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E5g  2i+2k+2m      {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
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+2l      {z(x2y2), xyz}, {z3(x2y2), xyz3} 
E3u  f+h+j+2l      {x(x2−3y2), y(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)} 
E4u  h+2j+2l       {z((x2y2)2−4x2y2), xyz(x2y2)} 
E5u  h+2j+2l       {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)} 

 Notes:

    α  The order of the D11d point group is 44, and the order of the principal axis (S22) is 22. The group has 14 irreducible representations.

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

    γ  The D11d point group is generated by two symmetry elements, S22 and either a perpendicular C2 or a vertical σd.
       Also, the group may be generated from any C2 plus any σd plane.
       The canonical choice, however, is to use redundant generators: C11, C2 and i.

    δ  The group contains one set of C2 symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set.
       Reversely, the single set of symmetry planes denoted σd contains the yz plane but not the xz plane.

    ε  The lowest nonvanishing multipole moment in D11d 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 just 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.

    θ  The regular hendecagon is the smallest regular polygon not constructible with ruler, compass and angle trisector.
       This is because 2*cos(2*π/11) has algebraic degree of five, being the solution of an irreducible quintic equation.
       Because this quintic equation is solvable, the value of cos(2*π/11) can be expressed using square and fifth roots and complex numbers.
       That algebraic form is, however, very complex and thus not shown here.

This Character Table for the D11d point group was created by Gernot Katzer.

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