Character table for the D58 point group

D58     E        2 C58    2 C29    2 C58^3  2 C29^2  2 C58^5  2 C29^3  2 C58^7  2 C29^4  2 C58^9  2 C29^5  2 C58^11 2 C29^6  2 C58^13 2 C29^7  2 C58^15 2 C29^8  2 C58^17 2 C29^9  2 C58^19 2 C29^10 2 C58^21 2 C29^11 2 C58^23 2 C29^12 2 C58^25 2 C29^13 2 C58^27 2 C29^14 C2       29 C2'   29 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  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  1.00000  1.00000  1.00000  1.00000  1.00000 -1.00000 -1.00000     ..T ..T ..... ......T ......... ..........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 -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 -1.00000  1.00000 -1.00000  1.00000 -1.00000 -1.00000  1.00000     ... ... ..... ....... ......... ........... .............
E1      2.00000  1.98828  1.95324  1.89531  1.81515  1.71371  1.59219  1.45199  1.29477  1.12237  0.93682  0.74028  0.53506  0.32356  0.10828 -0.10828 -0.32356 -0.53506 -0.74028 -0.93682 -1.12237 -1.29477 -1.45199 -1.59219 -1.71371 -1.81515 -1.89531 -1.95324 -1.98828 -2.00000  0.00000  0.00000     TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT.
E2      2.00000  1.95324  1.81515  1.59219  1.29477  0.93682  0.53506  0.10828 -0.32356 -0.74028 -1.12237 -1.45199 -1.71371 -1.89531 -1.98828 -1.98828 -1.89531 -1.71371 -1.45199 -1.12237 -0.74028 -0.32356  0.10828  0.53506  0.93682  1.29477  1.59219  1.81515  1.95324  2.00000  0.00000  0.00000     ... ... TT... ..TT... ....TT... ......TT... ........TT...
E3      2.00000  1.89531  1.59219  1.12237  0.53506 -0.10828 -0.74028 -1.29477 -1.71371 -1.95324 -1.98828 -1.81515 -1.45199 -0.93682 -0.32356  0.32356  0.93682  1.45199  1.81515  1.98828  1.95324  1.71371  1.29477  0.74028  0.10828 -0.53506 -1.12237 -1.59219 -1.89531 -2.00000  0.00000  0.00000     ... ... ..... TT..... ..TT..... ....TT..... ......TT.....
E4      2.00000  1.81515  1.29477  0.53506 -0.32356 -1.12237 -1.71371 -1.98828 -1.89531 -1.45199 -0.74028  0.10828  0.93682  1.59219  1.95324  1.95324  1.59219  0.93682  0.10828 -0.74028 -1.45199 -1.89531 -1.98828 -1.71371 -1.12237 -0.32356  0.53506  1.29477  1.81515  2.00000  0.00000  0.00000     ... ... ..... ....... TT....... ..TT....... ....TT.......
E5      2.00000  1.71371  0.93682 -0.10828 -1.12237 -1.81515 -1.98828 -1.59219 -0.74028  0.32356  1.29477  1.89531  1.95324  1.45199  0.53506 -0.53506 -1.45199 -1.95324 -1.89531 -1.29477 -0.32356  0.74028  1.59219  1.98828  1.81515  1.12237  0.10828 -0.93682 -1.71371 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... TT......... ..TT.........
E6      2.00000  1.59219  0.53506 -0.74028 -1.71371 -1.98828 -1.45199 -0.32356  0.93682  1.81515  1.95324  1.29477  0.10828 -1.12237 -1.89531 -1.89531 -1.12237  0.10828  1.29477  1.95324  1.81515  0.93682 -0.32356 -1.45199 -1.98828 -1.71371 -0.74028  0.53506  1.59219  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... TT...........
E7      2.00000  1.45199  0.10828 -1.29477 -1.98828 -1.59219 -0.32356  1.12237  1.95324  1.71371  0.53506 -0.93682 -1.89531 -1.81515 -0.74028  0.74028  1.81515  1.89531  0.93682 -0.53506 -1.71371 -1.95324 -1.12237  0.32356  1.59219  1.98828  1.29477 -0.10828 -1.45199 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E8      2.00000  1.29477 -0.32356 -1.71371 -1.89531 -0.74028  0.93682  1.95324  1.59219  0.10828 -1.45199 -1.98828 -1.12237  0.53506  1.81515  1.81515  0.53506 -1.12237 -1.98828 -1.45199  0.10828  1.59219  1.95324  0.93682 -0.74028 -1.89531 -1.71371 -0.32356  1.29477  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E9      2.00000  1.12237 -0.74028 -1.95324 -1.45199  0.32356  1.81515  1.71371  0.10828 -1.59219 -1.89531 -0.53506  1.29477  1.98828  0.93682 -0.93682 -1.98828 -1.29477  0.53506  1.89531  1.59219 -0.10828 -1.71371 -1.81515 -0.32356  1.45199  1.95324  0.74028 -1.12237 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E10     2.00000  0.93682 -1.12237 -1.98828 -0.74028  1.29477  1.95324  0.53506 -1.45199 -1.89531 -0.32356  1.59219  1.81515  0.10828 -1.71371 -1.71371  0.10828  1.81515  1.59219 -0.32356 -1.89531 -1.45199  0.53506  1.95324  1.29477 -0.74028 -1.98828 -1.12237  0.93682  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E11     2.00000  0.74028 -1.45199 -1.81515  0.10828  1.89531  1.29477 -0.93682 -1.98828 -0.53506  1.59219  1.71371 -0.32356 -1.95324 -1.12237  1.12237  1.95324  0.32356 -1.71371 -1.59219  0.53506  1.98828  0.93682 -1.29477 -1.89531 -0.10828  1.81515  1.45199 -0.74028 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E12     2.00000  0.53506 -1.71371 -1.45199  0.93682  1.95324  0.10828 -1.89531 -1.12237  1.29477  1.81515 -0.32356 -1.98828 -0.74028  1.59219  1.59219 -0.74028 -1.98828 -0.32356  1.81515  1.29477 -1.12237 -1.89531  0.10828  1.95324  0.93682 -1.45199 -1.71371  0.53506  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E13     2.00000  0.32356 -1.89531 -0.93682  1.59219  1.45199 -1.12237 -1.81515  0.53506  1.98828  0.10828 -1.95324 -0.74028  1.71371  1.29477 -1.29477 -1.71371  0.74028  1.95324 -0.10828 -1.98828 -0.53506  1.81515  1.12237 -1.45199 -1.59219  0.93682  1.89531 -0.32356 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E14     2.00000  0.10828 -1.98828 -0.32356  1.95324  0.53506 -1.89531 -0.74028  1.81515  0.93682 -1.71371 -1.12237  1.59219  1.29477 -1.45199 -1.45199  1.29477  1.59219 -1.12237 -1.71371  0.93682  1.81515 -0.74028 -1.89531  0.53506  1.95324 -0.32356 -1.98828  0.10828  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E15     2.00000 -0.10828 -1.98828  0.32356  1.95324 -0.53506 -1.89531  0.74028  1.81515 -0.93682 -1.71371  1.12237  1.59219 -1.29477 -1.45199  1.45199  1.29477 -1.59219 -1.12237  1.71371  0.93682 -1.81515 -0.74028  1.89531  0.53506 -1.95324 -0.32356  1.98828  0.10828 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E16     2.00000 -0.32356 -1.89531  0.93682  1.59219 -1.45199 -1.12237  1.81515  0.53506 -1.98828  0.10828  1.95324 -0.74028 -1.71371  1.29477  1.29477 -1.71371 -0.74028  1.95324  0.10828 -1.98828  0.53506  1.81515 -1.12237 -1.45199  1.59219  0.93682 -1.89531 -0.32356  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E17     2.00000 -0.53506 -1.71371  1.45199  0.93682 -1.95324  0.10828  1.89531 -1.12237 -1.29477  1.81515  0.32356 -1.98828  0.74028  1.59219 -1.59219 -0.74028  1.98828 -0.32356 -1.81515  1.29477  1.12237 -1.89531 -0.10828  1.95324 -0.93682 -1.45199  1.71371  0.53506 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E18     2.00000 -0.74028 -1.45199  1.81515  0.10828 -1.89531  1.29477  0.93682 -1.98828  0.53506  1.59219 -1.71371 -0.32356  1.95324 -1.12237 -1.12237  1.95324 -0.32356 -1.71371  1.59219  0.53506 -1.98828  0.93682  1.29477 -1.89531  0.10828  1.81515 -1.45199 -0.74028  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E19     2.00000 -0.93682 -1.12237  1.98828 -0.74028 -1.29477  1.95324 -0.53506 -1.45199  1.89531 -0.32356 -1.59219  1.81515 -0.10828 -1.71371  1.71371  0.10828 -1.81515  1.59219  0.32356 -1.89531  1.45199  0.53506 -1.95324  1.29477  0.74028 -1.98828  1.12237  0.93682 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E20     2.00000 -1.12237 -0.74028  1.95324 -1.45199 -0.32356  1.81515 -1.71371  0.10828  1.59219 -1.89531  0.53506  1.29477 -1.98828  0.93682  0.93682 -1.98828  1.29477  0.53506 -1.89531  1.59219  0.10828 -1.71371  1.81515 -0.32356 -1.45199  1.95324 -0.74028 -1.12237  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E21     2.00000 -1.29477 -0.32356  1.71371 -1.89531  0.74028  0.93682 -1.95324  1.59219 -0.10828 -1.45199  1.98828 -1.12237 -0.53506  1.81515 -1.81515  0.53506  1.12237 -1.98828  1.45199  0.10828 -1.59219  1.95324 -0.93682 -0.74028  1.89531 -1.71371  0.32356  1.29477 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E22     2.00000 -1.45199  0.10828  1.29477 -1.98828  1.59219 -0.32356 -1.12237  1.95324 -1.71371  0.53506  0.93682 -1.89531  1.81515 -0.74028 -0.74028  1.81515 -1.89531  0.93682  0.53506 -1.71371  1.95324 -1.12237 -0.32356  1.59219 -1.98828  1.29477  0.10828 -1.45199  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E23     2.00000 -1.59219  0.53506  0.74028 -1.71371  1.98828 -1.45199  0.32356  0.93682 -1.81515  1.95324 -1.29477  0.10828  1.12237 -1.89531  1.89531 -1.12237 -0.10828  1.29477 -1.95324  1.81515 -0.93682 -0.32356  1.45199 -1.98828  1.71371 -0.74028 -0.53506  1.59219 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E24     2.00000 -1.71371  0.93682  0.10828 -1.12237  1.81515 -1.98828  1.59219 -0.74028 -0.32356  1.29477 -1.89531  1.95324 -1.45199  0.53506  0.53506 -1.45199  1.95324 -1.89531  1.29477 -0.32356 -0.74028  1.59219 -1.98828  1.81515 -1.12237  0.10828  0.93682 -1.71371  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E25     2.00000 -1.81515  1.29477 -0.53506 -0.32356  1.12237 -1.71371  1.98828 -1.89531  1.45199 -0.74028 -0.10828  0.93682 -1.59219  1.95324 -1.95324  1.59219 -0.93682  0.10828  0.74028 -1.45199  1.89531 -1.98828  1.71371 -1.12237  0.32356  0.53506 -1.29477  1.81515 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E26     2.00000 -1.89531  1.59219 -1.12237  0.53506  0.10828 -0.74028  1.29477 -1.71371  1.95324 -1.98828  1.81515 -1.45199  0.93682 -0.32356 -0.32356  0.93682 -1.45199  1.81515 -1.98828  1.95324 -1.71371  1.29477 -0.74028  0.10828  0.53506 -1.12237  1.59219 -1.89531  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E27     2.00000 -1.95324  1.81515 -1.59219  1.29477 -0.93682  0.53506 -0.10828 -0.32356  0.74028 -1.12237  1.45199 -1.71371  1.89531 -1.98828  1.98828 -1.89531  1.71371 -1.45199  1.12237 -0.74028  0.32356  0.10828 -0.53506  0.93682 -1.29477  1.59219 -1.81515  1.95324 -2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............
E28     2.00000 -1.98828  1.95324 -1.89531  1.81515 -1.71371  1.59219 -1.45199  1.29477 -1.12237  0.93682 -0.74028  0.53506 -0.32356  0.10828  0.10828 -0.32356  0.53506 -0.74028  0.93682 -1.12237  1.29477 -1.45199  1.59219 -1.71371  1.81515 -1.89531  1.95324 -1.98828  2.00000  0.00000  0.00000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.988275914309 = 2*cos(2*π/58) = 2*cos(π/29)
                               1.953241111420 = 2*cos(4*π/58) = 2*cos(2*π/29)
                               1.895306342366 = 2*cos(6*π/58) = 2*cos(3*π/29)
                               1.815150839342 = 2*cos(8*π/58) = 2*cos(4*π/29)
                               1.713714352335 = 2*cos(10*π/58) = 2*cos(5*π/29)
                               1.592186131411 = 2*cos(12*π/58) = 2*cos(6*π/29)
                               1.451990983846 = 2*cos(14*π/58) = 2*cos(7*π/29)
                               1.294772569564 = 2*cos(16*π/58) = 2*cos(8*π/29)
                               1.122374130725 = 2*cos(18*π/58) = 2*cos(9*π/29)
                               0.936816881400 = 2*cos(20*π/58) = 2*cos(10*π/29)
                               0.740276310680 = 2*cos(22*π/58) = 2*cos(11*π/29)
                               0.535056677058 = 2*cos(24*π/58) = 2*cos(12*π/29)
                               0.323563993106 = 2*cos(26*π/58) = 2*cos(13*π/29)
                               0.108277817171 = 2*cos(28*π/58) = 2*cos(14*π/29)



 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 D58 point group is 116, and the order of the principal axis (C58) is 58. The group has 32 irreducible representations.

    β  The D58 point group is isomorphic to D29d, D29h and C58v.

    γ  The D58 point group is generated by two symmetry elements, C58 and a perpendicular C2 (or, non-canonically, C2).
       Also, the group may be generated from a C2 plus a C2 (some pairs will yield smaller groups, though; choosing a minimum angle is safe).

    δ  There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation).
       By convention, the set denoted as C2 has the x axis as a member, while the y axis is a member of the C2 set.

    ε  The lowest nonvanishing multipole moment in D58 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.
       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 D58 point group was created by Gernot Katzer.

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