Character table for the C_16v point group

C16v    E       2 C16   2 C8    2 C16^3 2 C4    2 C16^5 2 C8^3  2 C16^7 C2      8 sv    8 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     ... ..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     ..T ... ..... ....... ......... ........... .............
B1      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     ... ... ..... ....... ......... ........... .............
E1      2.0000  1.8477  1.4142  0.7653  0.0000 -0.7653 -1.4142 -1.8477 -2.0000  0.0000  0.0000     TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT.
E2      2.0000  1.4142  0.0000 -1.4142 -2.0000 -1.4142  0.0000  1.4142  2.0000  0.0000  0.0000     ... ... TT... ..TT... ....TT... ......TT... ........TT...
E3      2.0000  0.7653 -1.4142 -1.8477  0.0000  1.8477  1.4142 -0.7653 -2.0000  0.0000  0.0000     ... ... ..... TT..... ..TT..... ....TT..... ......TT.....
E4      2.0000  0.0000 -2.0000  0.0000  2.0000  0.0000 -2.0000  0.0000  2.0000  0.0000  0.0000     ... ... ..... ....... TT....... ..TT....... ....TT.......
E5      2.0000 -0.7653 -1.4142  1.8477  0.0000 -1.8477  1.4142  0.7653 -2.0000  0.0000  0.0000     ... ... ..... ....... ......... TT......... ..TT.........
E6      2.0000 -1.4142  0.0000  1.4142 -2.0000  1.4142  0.0000 -1.4142  2.0000  0.0000  0.0000     ... ... ..... ....... ......... ........... TT...........
E7      2.0000 -1.8477  1.4142 -0.7653  0.0000  0.7653 -1.4142  1.8477 -2.0000  0.0000  0.0000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.847759065023 = 2*cos(2*π/16) = 2*cos(π/8) = √2+√2
                               1.414213562373 = 2*cos(4*π/16) = 2*cos(π/4) = √2
                               0.765366864730 = 2*cos(6*π/16) = 2*cos(3*π/8) = √2−√2



 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 
B1   k+l+m 
B2   k+l+m 
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      {x2−y2, xy}, {z(x2−y2), xyz}, {z2(x2−y2), xyz2}, {z3(x2−y2), xyz3}, {z4(x2−y2), xyz4} 
E3   f+g+h+i+j+k+l+m        {x(x2−3y2), y(3x2−y2)}, {xz(x2−3y2), yz(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} 
E4   g+h+i+j+k+l+m          {(x2−y2)2−4x2y2, xy(x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} 
E5   h+i+j+k+l+m            {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} 
E6   i+j+k+l+2m             {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} 
E7   j+k+2l+2m 

 Notes:

    α  The order of the C16v point group is 32, and the order of the principal axis (C16) is 16. The group has 11 irreducible representations.

    β  The C16v point group is isomorphic to D8d and D16.

    γ  The C16v point group is generated by two symmetry elements, C16 and any σv (or, non-canonically, any σd).
       Also, the group may be generated from any σv plus any σd planes.

    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 contains both the xz and the yz planes.

    ε  The lowest nonvanishing multipole moment in C16v 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.

    θ  The point group corresponds to a constructible polygon, as the order of the principal axis is a product of any number
       of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an
       algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers.