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 {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+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} 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.
C59v | ||
C60v | ||
C61 | C61v | C61h D61 D61h D61d |
C62v | ||
C63v |
This Character Table for the C61v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.