Character table for the D59 point group

D59     E        2 C59    2 C59^2  2 C59^3  2 C59^4  2 C59^5  2 C59^6  2 C59^7  2 C59^8  2 C59^9  2 C59^10 2 C59^11 2 C59^12 2 C59^13 2 C59^14 2 C59^15 2 C59^16 2 C59^17 2 C59^18 2 C59^19 2 C59^20 2 C59^21 2 C59^22 2 C59^23 2 C59^24 2 C59^25 2 C59^26 2 C59^27 2 C59^28 2 C59^29 59 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     ... ... ....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     ..T ..T ..... ......T ......... ..........T .............
E1      2.00000  1.98867  1.95481  1.89880  1.82127  1.72311  1.60542  1.46955  1.31702  1.14957  0.96910  0.77765  0.57738  0.37058  0.15957 -0.05324 -0.26545 -0.47465 -0.67848 -0.87461 -1.06084 -1.23505 -1.39526 -1.53967 -1.66663 -1.77470 -1.86267 -1.92954 -1.97454 -1.99717  0.00000     TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT.
E2      2.00000  1.95481  1.82127  1.60542  1.31702  0.96910  0.57738  0.15957 -0.26545 -0.67848 -1.06084 -1.39526 -1.66663 -1.86267 -1.97454 -1.99717 -1.92954 -1.77470 -1.53967 -1.23505 -0.87461 -0.47465 -0.05324  0.37058  0.77765  1.14957  1.46955  1.72311  1.89880  1.98867  0.00000     ... ... TT... ..TT... ....TT... ......TT... ........TT...
E3      2.00000  1.89880  1.60542  1.14957  0.57738 -0.05324 -0.67848 -1.23505 -1.66663 -1.92954 -1.99717 -1.86267 -1.53967 -1.06084 -0.47465  0.15957  0.77765  1.31702  1.72311  1.95481  1.98867  1.82127  1.46955  0.96910  0.37058 -0.26545 -0.87461 -1.39526 -1.77470 -1.97454  0.00000     ... ... ..... TT..... ..TT..... ....TT..... ......TT.....
E4      2.00000  1.82127  1.31702  0.57738 -0.26545 -1.06084 -1.66663 -1.97454 -1.92954 -1.53967 -0.87461 -0.05324  0.77765  1.46955  1.89880  1.98867  1.72311  1.14957  0.37058 -0.47465 -1.23505 -1.77470 -1.99717 -1.86267 -1.39526 -0.67848  0.15957  0.96910  1.60542  1.95481  0.00000     ... ... ..... ....... TT....... ..TT....... ....TT.......
E5      2.00000  1.72311  0.96910 -0.05324 -1.06084 -1.77470 -1.99717 -1.66663 -0.87461  0.15957  1.14957  1.82127  1.98867  1.60542  0.77765 -0.26545 -1.23505 -1.86267 -1.97454 -1.53967 -0.67848  0.37058  1.31702  1.89880  1.95481  1.46955  0.57738 -0.47465 -1.39526 -1.92954  0.00000     ... ... ..... ....... ......... TT......... ..TT.........
E6      2.00000  1.60542  0.57738 -0.67848 -1.66663 -1.99717 -1.53967 -0.47465  0.77765  1.72311  1.98867  1.46955  0.37058 -0.87461 -1.77470 -1.97454 -1.39526 -0.26545  0.96910  1.82127  1.95481  1.31702  0.15957 -1.06084 -1.86267 -1.92954 -1.23505 -0.05324  1.14957  1.89880  0.00000     ... ... ..... ....... ......... ........... TT...........
E7      2.00000  1.46955  0.15957 -1.23505 -1.97454 -1.66663 -0.47465  0.96910  1.89880  1.82127  0.77765 -0.67848 -1.77470 -1.92954 -1.06084  0.37058  1.60542  1.98867  1.31702 -0.05324 -1.39526 -1.99717 -1.53967 -0.26545  1.14957  1.95481  1.72311  0.57738 -0.87461 -1.86267  0.00000     ... ... ..... ....... ......... ........... .............
E8      2.00000  1.31702 -0.26545 -1.66663 -1.92954 -0.87461  0.77765  1.89880  1.72311  0.37058 -1.23505 -1.99717 -1.39526  0.15957  1.60542  1.95481  0.96910 -0.67848 -1.86267 -1.77470 -0.47465  1.14957  1.98867  1.46955 -0.05324 -1.53967 -1.97454 -1.06084  0.57738  1.82127  0.00000     ... ... ..... ....... ......... ........... .............
E9      2.00000  1.14957 -0.67848 -1.92954 -1.53967  0.15957  1.72311  1.82127  0.37058 -1.39526 -1.97454 -0.87461  0.96910  1.98867  1.31702 -0.47465 -1.86267 -1.66663 -0.05324  1.60542  1.89880  0.57738 -1.23505 -1.99717 -1.06084  0.77765  1.95481  1.46955 -0.26545 -1.77470  0.00000     ... ... ..... ....... ......... ........... .............
E10     2.00000  0.96910 -1.06084 -1.99717 -0.87461  1.14957  1.98867  0.77765 -1.23505 -1.97454 -0.67848  1.31702  1.95481  0.57738 -1.39526 -1.92954 -0.47465  1.46955  1.89880  0.37058 -1.53967 -1.86267 -0.26545  1.60542  1.82127  0.15957 -1.66663 -1.77470 -0.05324  1.72311  0.00000     ... ... ..... ....... ......... ........... .............
E11     2.00000  0.77765 -1.39526 -1.86267 -0.05324  1.82127  1.46955 -0.67848 -1.99717 -0.87461  1.31702  1.89880  0.15957 -1.77470 -1.53967  0.57738  1.98867  0.96910 -1.23505 -1.92954 -0.26545  1.72311  1.60542 -0.47465 -1.97454 -1.06084  1.14957  1.95481  0.37058 -1.66663  0.00000     ... ... ..... ....... ......... ........... .............
E12     2.00000  0.57738 -1.66663 -1.53967  0.77765  1.98867  0.37058 -1.77470 -1.39526  0.96910  1.95481  0.15957 -1.86267 -1.23505  1.14957  1.89880 -0.05324 -1.92954 -1.06084  1.31702  1.82127 -0.26545 -1.97454 -0.87461  1.46955  1.72311 -0.47465 -1.99717 -0.67848  1.60542  0.00000     ... ... ..... ....... ......... ........... .............
E13     2.00000  0.37058 -1.86267 -1.06084  1.46955  1.60542 -0.87461 -1.92954  0.15957  1.98867  0.57738 -1.77470 -1.23505  1.31702  1.72311 -0.67848 -1.97454 -0.05324  1.95481  0.77765 -1.66663 -1.39526  1.14957  1.82127 -0.47465 -1.99717 -0.26545  1.89880  0.96910 -1.53967  0.00000     ... ... ..... ....... ......... ........... .............
E14     2.00000  0.15957 -1.97454 -0.47465  1.89880  0.77765 -1.77470 -1.06084  1.60542  1.31702 -1.39526 -1.53967  1.14957  1.72311 -0.87461 -1.86267  0.57738  1.95481 -0.26545 -1.99717 -0.05324  1.98867  0.37058 -1.92954 -0.67848  1.82127  0.96910 -1.66663 -1.23505  1.46955  0.00000     ... ... ..... ....... ......... ........... .............
E15     2.00000 -0.05324 -1.99717  0.15957  1.98867 -0.26545 -1.97454  0.37058  1.95481 -0.47465 -1.92954  0.57738  1.89880 -0.67848 -1.86267  0.77765  1.82127 -0.87461 -1.77470  0.96910  1.72311 -1.06084 -1.66663  1.14957  1.60542 -1.23505 -1.53967  1.31702  1.46955 -1.39526  0.00000     ... ... ..... ....... ......... ........... .............
E16     2.00000 -0.26545 -1.92954  0.77765  1.72311 -1.23505 -1.39526  1.60542  0.96910 -1.86267 -0.47465  1.98867 -0.05324 -1.97454  0.57738  1.82127 -1.06084 -1.53967  1.46955  1.14957 -1.77470 -0.67848  1.95481  0.15957 -1.99717  0.37058  1.89880 -0.87461 -1.66663  1.31702  0.00000     ... ... ..... ....... ......... ........... .............
E17     2.00000 -0.47465 -1.77470  1.31702  1.14957 -1.86267 -0.26545  1.98867 -0.67848 -1.66663  1.46955  0.96910 -1.92954 -0.05324  1.95481 -0.87461 -1.53967  1.60542  0.77765 -1.97454  0.15957  1.89880 -1.06084 -1.39526  1.72311  0.57738 -1.99717  0.37058  1.82127 -1.23505  0.00000     ... ... ..... ....... ......... ........... .............
E18     2.00000 -0.67848 -1.53967  1.72311  0.37058 -1.97454  0.96910  1.31702 -1.86267 -0.05324  1.89880 -1.23505 -1.06084  1.95481 -0.26545 -1.77470  1.46955  0.77765 -1.99717  0.57738  1.60542 -1.66663 -0.47465  1.98867 -0.87461 -1.39526  1.82127  0.15957 -1.92954  1.14957  0.00000     ... ... ..... ....... ......... ........... .............
E19     2.00000 -0.87461 -1.23505  1.95481 -0.47465 -1.53967  1.82127 -0.05324 -1.77470  1.60542  0.37058 -1.92954  1.31702  0.77765 -1.99717  0.96910  1.14957 -1.97454  0.57738  1.46955 -1.86267  0.15957  1.72311 -1.66663 -0.26545  1.89880 -1.39526 -0.67848  1.98867 -1.06084  0.00000     ... ... ..... ....... ......... ........... .............
E20     2.00000 -1.06084 -0.87461  1.98867 -1.23505 -0.67848  1.95481 -1.39526 -0.47465  1.89880 -1.53967 -0.26545  1.82127 -1.66663 -0.05324  1.72311 -1.77470  0.15957  1.60542 -1.86267  0.37058  1.46955 -1.92954  0.57738  1.31702 -1.97454  0.77765  1.14957 -1.99717  0.96910  0.00000     ... ... ..... ....... ......... ........... .............
E21     2.00000 -1.23505 -0.47465  1.82127 -1.77470  0.37058  1.31702 -1.99717  1.14957  0.57738 -1.86267  1.72311 -0.26545 -1.39526  1.98867 -1.06084 -0.67848  1.89880 -1.66663  0.15957  1.46955 -1.97454  0.96910  0.77765 -1.92954  1.60542 -0.05324 -1.53967  1.95481 -0.87461  0.00000     ... ... ..... ....... ......... ........... .............
E22     2.00000 -1.39526 -0.05324  1.46955 -1.99717  1.31702  0.15957 -1.53967  1.98867 -1.23505 -0.26545  1.60542 -1.97454  1.14957  0.37058 -1.66663  1.95481 -1.06084 -0.47465  1.72311 -1.92954  0.96910  0.57738 -1.77470  1.89880 -0.87461 -0.67848  1.82127 -1.86267  0.77765  0.00000     ... ... ..... ....... ......... ........... .............
E23     2.00000 -1.53967  0.37058  0.96910 -1.86267  1.89880 -1.06084 -0.26545  1.46955 -1.99717  1.60542 -0.47465 -0.87461  1.82127 -1.92954  1.14957  0.15957 -1.39526  1.98867 -1.66663  0.57738  0.77765 -1.77470  1.95481 -1.23505 -0.05324  1.31702 -1.97454  1.72311 -0.67848  0.00000     ... ... ..... ....... ......... ........... .............
E24     2.00000 -1.66663  0.77765  0.37058 -1.39526  1.95481 -1.86267  1.14957 -0.05324 -1.06084  1.82127 -1.97454  1.46955 -0.47465 -0.67848  1.60542 -1.99717  1.72311 -0.87461 -0.26545  1.31702 -1.92954  1.89880 -1.23505  0.15957  0.96910 -1.77470  1.98867 -1.53967  0.57738  0.00000     ... ... ..... ....... ......... ........... .............
E25     2.00000 -1.77470  1.14957 -0.26545 -0.67848  1.46955 -1.92954  1.95481 -1.53967  0.77765  0.15957 -1.06084  1.72311 -1.99717  1.82127 -1.23505  0.37058  0.57738 -1.39526  1.89880 -1.97454  1.60542 -0.87461 -0.05324  0.96910 -1.66663  1.98867 -1.86267  1.31702 -0.47465  0.00000     ... ... ..... ....... ......... ........... .............
E26     2.00000 -1.86267  1.46955 -0.87461  0.15957  0.57738 -1.23505  1.72311 -1.97454  1.95481 -1.66663  1.14957 -0.47465 -0.26545  0.96910 -1.53967  1.89880 -1.99717  1.82127 -1.39526  0.77765 -0.05324 -0.67848  1.31702 -1.77470  1.98867 -1.92954  1.60542 -1.06084  0.37058  0.00000     ... ... ..... ....... ......... ........... .............
E27     2.00000 -1.92954  1.72311 -1.39526  0.96910 -0.47465 -0.05324  0.57738 -1.06084  1.46955 -1.77470  1.95481 -1.99717  1.89880 -1.66663  1.31702 -0.87461  0.37058  0.15957 -0.67848  1.14957 -1.53967  1.82127 -1.97454  1.98867 -1.86267  1.60542 -1.23505  0.77765 -0.26545  0.00000     ... ... ..... ....... ......... ........... .............
E28     2.00000 -1.97454  1.89880 -1.77470  1.60542 -1.39526  1.14957 -0.87461  0.57738 -0.26545 -0.05324  0.37058 -0.67848  0.96910 -1.23505  1.46955 -1.66663  1.82127 -1.92954  1.98867 -1.99717  1.95481 -1.86267  1.72311 -1.53967  1.31702 -1.06084  0.77765 -0.47465  0.15957  0.00000     ... ... ..... ....... ......... ........... .............
E29     2.00000 -1.99717  1.98867 -1.97454  1.95481 -1.92954  1.89880 -1.86267  1.82127 -1.77470  1.72311 -1.66663  1.60542 -1.53967  1.46955 -1.39526  1.31702 -1.23505  1.14957 -1.06084  0.96910 -0.87461  0.77765 -0.67848  0.57738 -0.47465  0.37058 -0.26545  0.15957 -0.05324  0.00000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.988669600420 = 2*cos(2*π/59)
                               1.954806779636 = 2*cos(4*π/59)
                               1.898795216937 = 2*cos(6*π/59)
                               1.821269545710 = 2*cos(8*π/59)
                               1.723108162788 = 2*cos(10*π/59)
                               1.605423275862 = 2*cos(12*π/59)
                               1.469548301726 = 2*cos(14*π/59)
                               1.317022758130 = 2*cos(16*π/59)
                               1.149574820429 = 2*cos(18*π/59)
                               0.969101740665 = 2*cos(20*π/59)
                               0.777648350947 = 2*cos(22*π/59)
                               0.577383894679 = 2*cos(24*π/59)
                               0.370577448174 = 2*cos(26*π/59)
                               0.159572211106 = 2*cos(28*π/59)
                              -0.053241042876 = 2*cos(30*π/59)
                              -0.265451054567 = 2*cos(32*π/59)
                              -0.474653399742 = 2*cos(34*π/59)
                              -0.678477732236 = 2*cos(36*π/59)
                              -0.874614640918 = 2*cos(38*π/59)
                              -1.060841816239 = 2*cos(40*π/59)
                              -1.235049229892 = 2*cos(42*π/59)
                              -1.395263042270 = 2*cos(44*π/59)
                              -1.539667966860 = 2*cos(46*π/59)
                              -1.666627838165 = 2*cos(48*π/59)
                              -1.774704150113 = 2*cos(50*π/59)
                              -1.862672354905 = 2*cos(52*π/59)
                              -1.929535737629 = 2*cos(54*π/59)
                              -1.974536709443 = 2*cos(56*π/59)
                              -1.997165391354 = 2*cos(58*π/59)



 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 D59 point group is 118, and the order of the principal axis (C59) is 59. The group has 31 irreducible representations.

    β  The D59 point group is isomorphic to C59v.

    γ  The D59 point group is generated by two symmetry elements, C59 and a perpendicular C2.
       Also, the group may be generated from any two C2 axes.

    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 D59 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 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.

This Character Table for the D59 point group was created by Gernot Katzer.

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