C59v 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 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 ... ..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 ..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 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 C59v point group is 118, and the order of the principal axis (C59) is 59. The group has 31 irreducible representations. β The C59v point group is isomorphic to D59. γ The C59v point group is generated by two symmetry elements, C59 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 C59v 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.
| C57v | ||
| C58v | ||
| C59 | C59v | C59h D59 D59h D59d |
| C60v | ||
| C61v |
This Character Table for the C59v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.