Character table for the C61v point group

C61v    E        2 C61    2 C61^2  2 C61^3  2 C61^4  2 C61^5  2 C61^6  2 C61^7  2 C61^8  2 C61^9  2 C61^10 2 C61^11 2 C61^12 2 C61^13 2 C61^14 2 C61^15 2 C61^16 2 C61^17 2 C61^18 2 C61^19 2 C61^20 2 C61^21 2 C61^22 2 C61^23 2 C61^24 2 C61^25 2 C61^26 2 C61^27 2 C61^28 2 C61^29 2 C61^30 61 sv       <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 ........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 ... ..... ....... ......... ........... .............
E1      2.00000  1.98940  1.95771  1.90527  1.83263  1.74057  1.63006  1.50226  1.35855  1.20043  1.02959  0.84783  0.65708  0.45938  0.25680  0.05150 -0.15435 -0.35856 -0.55897 -0.75346 -0.93995 -1.11649 -1.28119 -1.43230 -1.56824 -1.68755 -1.78897 -1.87143 -1.93406 -1.97618 -1.99735  0.00000     TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT.
E2      2.00000  1.95771  1.83263  1.63006  1.35855  1.02959  0.65708  0.25680 -0.15435 -0.55897 -0.93995 -1.28119 -1.56824 -1.78897 -1.93406 -1.99735 -1.97618 -1.87143 -1.68755 -1.43230 -1.11649 -0.75346 -0.35856  0.05150  0.45938  0.84783  1.20043  1.50226  1.74057  1.90527  1.98940  0.00000     ... ... TT... ..TT... ....TT... ......TT... ........TT...
E3      2.00000  1.90527  1.63006  1.20043  0.65708  0.05150 -0.55897 -1.11649 -1.56824 -1.87143 -1.99735 -1.93406 -1.68755 -1.28119 -0.75346 -0.15435  0.45938  1.02959  1.50226  1.83263  1.98940  1.95771  1.74057  1.35855  0.84783  0.25680 -0.35856 -0.93995 -1.43230 -1.78897 -1.97618  0.00000     ... ... ..... TT..... ..TT..... ....TT..... ......TT.....
E4      2.00000  1.83263  1.35855  0.65708 -0.15435 -0.93995 -1.56824 -1.93406 -1.97618 -1.68755 -1.11649 -0.35856  0.45938  1.20043  1.74057  1.98940  1.90527  1.50226  0.84783  0.05150 -0.75346 -1.43230 -1.87143 -1.99735 -1.78897 -1.28119 -0.55897  0.25680  1.02959  1.63006  1.95771  0.00000     ... ... ..... ....... TT....... ..TT....... ....TT.......
E5      2.00000  1.74057  1.02959  0.05150 -0.93995 -1.68755 -1.99735 -1.78897 -1.11649 -0.15435  0.84783  1.63006  1.98940  1.83263  1.20043  0.25680 -0.75346 -1.56824 -1.97618 -1.87143 -1.28119 -0.35856  0.65708  1.50226  1.95771  1.90527  1.35855  0.45938 -0.55897 -1.43230 -1.93406  0.00000     ... ... ..... ....... ......... TT......... ..TT.........
E6      2.00000  1.63006  0.65708 -0.55897 -1.56824 -1.99735 -1.68755 -0.75346  0.45938  1.50226  1.98940  1.74057  0.84783 -0.35856 -1.43230 -1.97618 -1.78897 -0.93995  0.25680  1.35855  1.95771  1.83263  1.02959 -0.15435 -1.28119 -1.93406 -1.87143 -1.11649  0.05150  1.20043  1.90527  0.00000     ... ... ..... ....... ......... ........... TT...........
E7      2.00000  1.50226  0.25680 -1.11649 -1.93406 -1.78897 -0.75346  0.65708  1.74057  1.95771  1.20043 -0.15435 -1.43230 -1.99735 -1.56824 -0.35856  1.02959  1.90527  1.83263  0.84783 -0.55897 -1.68755 -1.97618 -1.28119  0.05150  1.35855  1.98940  1.63006  0.45938 -0.93995 -1.87143  0.00000     ... ... ..... ....... ......... ........... .............
E8      2.00000  1.35855 -0.15435 -1.56824 -1.97618 -1.11649  0.45938  1.74057  1.90527  0.84783 -0.75346 -1.87143 -1.78897 -0.55897  1.02959  1.95771  1.63006  0.25680 -1.28119 -1.99735 -1.43230  0.05150  1.50226  1.98940  1.20043 -0.35856 -1.68755 -1.93406 -0.93995  0.65708  1.83263  0.00000     ... ... ..... ....... ......... ........... .............
E9      2.00000  1.20043 -0.55897 -1.87143 -1.68755 -0.15435  1.50226  1.95771  0.84783 -0.93995 -1.97618 -1.43230  0.25680  1.74057  1.83263  0.45938 -1.28119 -1.99735 -1.11649  0.65708  1.90527  1.63006  0.05150 -1.56824 -1.93406 -0.75346  1.02959  1.98940  1.35855 -0.35856 -1.78897  0.00000     ... ... ..... ....... ......... ........... .............
E10     2.00000  1.02959 -0.93995 -1.99735 -1.11649  0.84783  1.98940  1.20043 -0.75346 -1.97618 -1.28119  0.65708  1.95771  1.35855 -0.55897 -1.93406 -1.43230  0.45938  1.90527  1.50226 -0.35856 -1.87143 -1.56824  0.25680  1.83263  1.63006 -0.15435 -1.78897 -1.68755  0.05150  1.74057  0.00000     ... ... ..... ....... ......... ........... .............
E11     2.00000  0.84783 -1.28119 -1.93406 -0.35856  1.63006  1.74057 -0.15435 -1.87143 -1.43230  0.65708  1.98940  1.02959 -1.11649 -1.97618 -0.55897  1.50226  1.83263  0.05150 -1.78897 -1.56824  0.45938  1.95771  1.20043 -0.93995 -1.99735 -0.75346  1.35855  1.90527  0.25680 -1.68755  0.00000     ... ... ..... ....... ......... ........... .............
E12     2.00000  0.65708 -1.56824 -1.68755  0.45938  1.98940  0.84783 -1.43230 -1.78897  0.25680  1.95771  1.02959 -1.28119 -1.87143  0.05150  1.90527  1.20043 -1.11649 -1.93406 -0.15435  1.83263  1.35855 -0.93995 -1.97618 -0.35856  1.74057  1.50226 -0.75346 -1.99735 -0.55897  1.63006  0.00000     ... ... ..... ....... ......... ........... .............
E13     2.00000  0.45938 -1.78897 -1.28119  1.20043  1.83263 -0.35856 -1.99735 -0.55897  1.74057  1.35855 -1.11649 -1.87143  0.25680  1.98940  0.65708 -1.68755 -1.43230  1.02959  1.90527 -0.15435 -1.97618 -0.75346  1.63006  1.50226 -0.93995 -1.93406  0.05150  1.95771  0.84783 -1.56824  0.00000     ... ... ..... ....... ......... ........... .............
E14     2.00000  0.25680 -1.93406 -0.75346  1.74057  1.20043 -1.43230 -1.56824  1.02959  1.83263 -0.55897 -1.97618  0.05150  1.98940  0.45938 -1.87143 -0.93995  1.63006  1.35855 -1.28119 -1.68755  0.84783  1.90527 -0.35856 -1.99735 -0.15435  1.95771  0.65708 -1.78897 -1.11649  1.50226  0.00000     ... ... ..... ....... ......... ........... .............
E15     2.00000  0.05150 -1.99735 -0.15435  1.98940  0.25680 -1.97618 -0.35856  1.95771  0.45938 -1.93406 -0.55897  1.90527  0.65708 -1.87143 -0.75346  1.83263  0.84783 -1.78897 -0.93995  1.74057  1.02959 -1.68755 -1.11649  1.63006  1.20043 -1.56824 -1.28119  1.50226  1.35855 -1.43230  0.00000     ... ... ..... ....... ......... ........... .............
E16     2.00000 -0.15435 -1.97618  0.45938  1.90527 -0.75346 -1.78897  1.02959  1.63006 -1.28119 -1.43230  1.50226  1.20043 -1.68755 -0.93995  1.83263  0.65708 -1.93406 -0.35856  1.98940  0.05150 -1.99735  0.25680  1.95771 -0.55897 -1.87143  0.84783  1.74057 -1.11649 -1.56824  1.35855  0.00000     ... ... ..... ....... ......... ........... .............
E17     2.00000 -0.35856 -1.87143  1.02959  1.50226 -1.56824 -0.93995  1.90527  0.25680 -1.99735  0.45938  1.83263 -1.11649 -1.43230  1.63006  0.84783 -1.93406 -0.15435  1.98940 -0.55897 -1.78897  1.20043  1.35855 -1.68755 -0.75346  1.95771  0.05150 -1.97618  0.65708  1.74057 -1.28119  0.00000     ... ... ..... ....... ......... ........... .............
E18     2.00000 -0.55897 -1.68755  1.50226  0.84783 -1.97618  0.25680  1.83263 -1.28119 -1.11649  1.90527  0.05150 -1.93406  1.02959  1.35855 -1.78897 -0.35856  1.98940 -0.75346 -1.56824  1.63006  0.65708 -1.99735  0.45938  1.74057 -1.43230 -0.93995  1.95771 -0.15435 -1.87143  1.20043  0.00000     ... ... ..... ....... ......... ........... .............
E19     2.00000 -0.75346 -1.43230  1.83263  0.05150 -1.87143  1.35855  0.84783 -1.99735  0.65708  1.50226 -1.78897 -0.15435  1.90527 -1.28119 -0.93995  1.98940 -0.55897 -1.56824  1.74057  0.25680 -1.93406  1.20043  1.02959 -1.97618  0.45938  1.63006 -1.68755 -0.35856  1.95771 -1.11649  0.00000     ... ... ..... ....... ......... ........... .............
E20     2.00000 -0.93995 -1.11649  1.98940 -0.75346 -1.28119  1.95771 -0.55897 -1.43230  1.90527 -0.35856 -1.56824  1.83263 -0.15435 -1.68755  1.74057  0.05150 -1.78897  1.63006  0.25680 -1.87143  1.50226  0.45938 -1.93406  1.35855  0.65708 -1.97618  1.20043  0.84783 -1.99735  1.02959  0.00000     ... ... ..... ....... ......... ........... .............
E21     2.00000 -1.11649 -0.75346  1.95771 -1.43230 -0.35856  1.83263 -1.68755  0.05150  1.63006 -1.87143  0.45938  1.35855 -1.97618  0.84783  1.02959 -1.99735  1.20043  0.65708 -1.93406  1.50226  0.25680 -1.78897  1.74057 -0.15435 -1.56824  1.90527 -0.55897 -1.28119  1.98940 -0.93995  0.00000     ... ... ..... ....... ......... ........... .............
E22     2.00000 -1.28119 -0.35856  1.74057 -1.87143  0.65708  1.02959 -1.97618  1.50226  0.05150 -1.56824  1.95771 -0.93995 -0.75346  1.90527 -1.68755  0.25680  1.35855 -1.99735  1.20043  0.45938 -1.78897  1.83263 -0.55897 -1.11649  1.98940 -1.43230 -0.15435  1.63006 -1.93406  0.84783  0.00000     ... ... ..... ....... ......... ........... .............
E23     2.00000 -1.43230  0.05150  1.35855 -1.99735  1.50226 -0.15435 -1.28119  1.98940 -1.56824  0.25680  1.20043 -1.97618  1.63006 -0.35856 -1.11649  1.95771 -1.68755  0.45938  1.02959 -1.93406  1.74057 -0.55897 -0.93995  1.90527 -1.78897  0.65708  0.84783 -1.87143  1.83263 -0.75346  0.00000     ... ... ..... ....... ......... ........... .............
E24     2.00000 -1.56824  0.45938  0.84783 -1.78897  1.95771 -1.28119  0.05150  1.20043 -1.93406  1.83263 -0.93995 -0.35856  1.50226 -1.99735  1.63006 -0.55897 -0.75346  1.74057 -1.97618  1.35855 -0.15435 -1.11649  1.90527 -1.87143  1.02959  0.25680 -1.43230  1.98940 -1.68755  0.65708  0.00000     ... ... ..... ....... ......... ........... .............
E25     2.00000 -1.68755  0.84783  0.25680 -1.28119  1.90527 -1.93406  1.35855 -0.35856 -0.75346  1.63006 -1.99735  1.74057 -0.93995 -0.15435  1.20043 -1.87143  1.95771 -1.43230  0.45938  0.65708 -1.56824  1.98940 -1.78897  1.02959  0.05150 -1.11649  1.83263 -1.97618  1.50226 -0.55897  0.00000     ... ... ..... ....... ......... ........... .............
E26     2.00000 -1.78897  1.20043 -0.35856 -0.55897  1.35855 -1.87143  1.98940 -1.68755  1.02959 -0.15435 -0.75346  1.50226 -1.93406  1.95771 -1.56824  0.84783  0.05150 -0.93995  1.63006 -1.97618  1.90527 -1.43230  0.65708  0.25680 -1.11649  1.74057 -1.99735  1.83263 -1.28119  0.45938  0.00000     ... ... ..... ....... ......... ........... .............
E27     2.00000 -1.87143  1.50226 -0.93995  0.25680  0.45938 -1.11649  1.63006 -1.93406  1.98940 -1.78897  1.35855 -0.75346  0.05150  0.65708 -1.28119  1.74057 -1.97618  1.95771 -1.68755  1.20043 -0.55897 -0.15435  0.84783 -1.43230  1.83263 -1.99735  1.90527 -1.56824  1.02959 -0.35856  0.00000     ... ... ..... ....... ......... ........... .............
E28     2.00000 -1.93406  1.74057 -1.43230  1.02959 -0.55897  0.05150  0.45938 -0.93995  1.35855 -1.68755  1.90527 -1.99735  1.95771 -1.78897  1.50226 -1.11649  0.65708 -0.15435 -0.35856  0.84783 -1.28119  1.63006 -1.87143  1.98940 -1.97618  1.83263 -1.56824  1.20043 -0.75346  0.25680  0.00000     ... ... ..... ....... ......... ........... .............
E29     2.00000 -1.97618  1.90527 -1.78897  1.63006 -1.43230  1.20043 -0.93995  0.65708 -0.35856  0.05150  0.25680 -0.55897  0.84783 -1.11649  1.35855 -1.56824  1.74057 -1.87143  1.95771 -1.99735  1.98940 -1.93406  1.83263 -1.68755  1.50226 -1.28119  1.02959 -0.75346  0.45938 -0.15435  0.00000     ... ... ..... ....... ......... ........... .............
E30     2.00000 -1.99735  1.98940 -1.97618  1.95771 -1.93406  1.90527 -1.87143  1.83263 -1.78897  1.74057 -1.68755  1.63006 -1.56824  1.50226 -1.43230  1.35855 -1.28119  1.20043 -1.11649  1.02959 -0.93995  0.84783 -0.75346  0.65708 -0.55897  0.45938 -0.35856  0.25680 -0.15435  0.05150  0.00000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.989399751229 = 2*cos(2*π/61)
                               1.957711370191 = 2*cos(4*π/61)
                               1.905270761607 = 2*cos(6*π/61)
                               1.832633808974 = 2*cos(8*π/61)
                               1.740570482060 = 2*cos(10*π/61)
                               1.630056675034 = 2*cos(12*π/61)
                               1.502263861741 = 2*cos(14*π/61)
                               1.358546677795 = 2*cos(16*π/61)
                               1.200428561097 = 2*cos(18*π/61)
                               1.029585603020 = 2*cos(20*π/61)
                               0.847828781420 = 2*cos(22*π/61)
                               0.657084763822 = 2*cos(24*π/61)
                               0.459375484264 = 2*cos(26*π/61)
                               0.256796710293 = 2*cos(28*π/61)
                               0.051495827310 = 2*cos(30*π/61)
                              -0.154350924253 = 2*cos(32*π/61)
                              -0.358561517621 = 2*cos(34*π/61)
                              -0.558971269703 = 2*cos(36*π/61)
                              -0.753455787270 = 2*cos(38*π/61)
                              -0.939953486055 = 2*cos(40*π/61)
                              -1.116487444054 = 2*cos(42*π/61)
                              -1.281186357396 = 2*cos(44*π/61)
                              -1.432304376629 = 2*cos(46*π/61)
                              -1.568239613153 = 2*cos(48*π/61)
                              -1.687551119646 = 2*cos(50*π/61)
                              -1.788974164458 = 2*cos(52*π/61)
                              -1.871433638081 = 2*cos(54*π/61)
                              -1.934055449583 = 2*cos(56*π/61)
                              -1.976175792182 = 2*cos(58*π/61)
                              -1.997348179770 = 2*cos(60*π/61)



 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 
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 C61v point group is 122, and the order of the principal axis (C61) is 61. The group has 32 irreducible representations.

    β  The C61v point group is isomorphic to D61.

    γ  The C61v point group is generated by two symmetry elements, C61 and any σv.
       Also, the group may be generated from any two σv planes.

    δ  The group contains one set of symmetry planes σv intersecting in the principal (z) axis. The xz plane (but not the yz plane) is a member of that set.

    ε  The lowest nonvanishing multipole moment in C61v 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 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 C61v point group was created by Gernot Katzer.

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