Character table for the D21 point group

D21     E        2 C21    2 C21^2  2 C7     2 C21^4  2 C21^5  2 C7^2   2 C3     2 C21^8  2 C7^3   2 C21^10 21 C2'      <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     ... ... ....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     ..T ..T ..... ......T ......... ..........T .............
E1      2.00000  1.91115  1.65248  1.24698  0.73068  0.14946 -0.44504 -1.00000 -1.46610 -1.80194 -1.97766  0.00000     TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT.
E2      2.00000  1.65248  0.73068 -0.44504 -1.46610 -1.97766 -1.80194 -1.00000  0.14946  1.24698  1.91115  0.00000     ... ... TT... ..TT... ....TT... ......TT... ........TT...
E3      2.00000  1.24698 -0.44504 -1.80194 -1.80194 -0.44504  1.24698  2.00000  1.24698 -0.44504 -1.80194  0.00000     ... ... ..... TT..... ..TT..... ....TT..... ......TT.....
E4      2.00000  0.73068 -1.46610 -1.80194  0.14946  1.91115  1.24698 -1.00000 -1.97766 -0.44504  1.65248  0.00000     ... ... ..... ....... TT....... ..TT....... ....TT.......
E5      2.00000  0.14946 -1.97766 -0.44504  1.91115  0.73068 -1.80194 -1.00000  1.65248  1.24698 -1.46610  0.00000     ... ... ..... ....... ......... TT......... ..TT.........
E6      2.00000 -0.44504 -1.80194  1.24698  1.24698 -1.80194 -0.44504  2.00000 -0.44504 -1.80194  1.24698  0.00000     ... ... ..... ....... ......... ........... TT...........
E7      2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000  0.00000     ... ... ..... ....... ......... ........... .............
E8      2.00000 -1.46610  0.14946  1.24698 -1.97766  1.65248 -0.44504 -1.00000  1.91115 -1.80194  0.73068  0.00000     ... ... ..... ....... ......... ........... .............
E9      2.00000 -1.80194  1.24698 -0.44504 -0.44504  1.24698 -1.80194  2.00000 -1.80194  1.24698 -0.44504  0.00000     ... ... ..... ....... ......... ........... .............
E10     2.00000 -1.97766  1.91115 -1.80194  1.65248 -1.46610  1.24698 -1.00000  0.73068 -0.44504  0.14946  0.00000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.911145611572 = 2*cos(2*π/21)
                               1.652477548632 = 2*cos(4*π/21)
                               1.246979603717 = 2*cos(6*π/21) = 2*cos(2*π/7)
                               0.730682048733 = 2*cos(8*π/21)
                               0.149460187173 = 2*cos(10*π/21)
                              -0.445041867913 = 2*cos(12*π/21) = 2*cos(4*π/7)
                              -1.466103743660 = 2*cos(16*π/21)
                              -1.801937735805 = 2*cos(18*π/21) = 2*cos(6*π/7)
                              -1.977661652450 = 2*cos(20*π/21)



 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      {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 D21 point group is 42, and the order of the principal axis (C21) is 21. The group has 12 irreducible representations.

    β  The D21 point group is isomorphic to C21v.

    γ  The D21 point group is generated by two symmetry elements, C21 and a perpendicular C2.
       Also, the group may be generated from two C2 axes (some pairs will yield smaller groups, though; choosing a minimum angle is safe).

    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.

    ε  The lowest nonvanishing multipole moment in D21 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.

    ι  The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible
       if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given
       by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three.
       All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.

This Character Table for the D21 point group was created by Gernot Katzer.

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