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E35 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 0.00000 ... ... ..... ....... ......... ........... ............. E36 2.00000 -1.95853 1.83583 -1.63700 1.37028 -1.04674 0.67978 -0.28463 -0.12232 0.52420 -0.90434 1.24698 -1.53790 1.76504 -1.91899 1.99335 -1.98504 1.89441 -1.72521 1.48447 -1.18216 0.83083 -0.44504 0.04080 0.36514 -0.75593 1.11538 -1.42856 1.68251 -1.86667 1.97342 -1.99834 1.94037 -1.80194 1.58877 -1.30972 0.97635 -0.60249 0.20365 0.00000 ... ... ..... ....... ......... ........... ............. E37 2.00000 -1.98504 1.94037 -1.86667 1.76504 -1.63700 1.48447 -1.30972 1.11538 -0.90434 0.67978 -0.44504 0.20365 0.04080 -0.28463 0.52420 -0.75593 0.97635 -1.18216 1.37028 -1.53790 1.68251 -1.80194 1.89441 -1.95853 1.99335 -1.99834 1.97342 -1.91899 1.83583 -1.72521 1.58877 -1.42856 1.24698 -1.04674 0.83083 -0.60249 0.36514 -0.12232 0.00000 ... ... ..... ....... ......... ........... ............. E38 2.00000 -1.99834 1.99335 -1.98504 1.97342 -1.95853 1.94037 -1.91899 1.89441 -1.86667 1.83583 -1.80194 1.76504 -1.72521 1.68251 -1.63700 1.58877 -1.53790 1.48447 -1.42856 1.37028 -1.30972 1.24698 -1.18216 1.11538 -1.04674 0.97635 -0.90434 0.83083 -0.75593 0.67978 -0.60249 0.52420 -0.44504 0.36514 -0.28463 0.20365 -0.12232 0.04080 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.993345164988 = 2*cos(2*π/77) 1.973424946783 = 2*cos(4*π/77) 1.940371911148 = 2*cos(6*π/77) 1.894406020584 = 2*cos(8*π/77) 1.835833170508 = 2*cos(10*π/77) 1.765043153573 = 2*cos(12*π/77) 1.682507065662 = 2*cos(14*π/77) = 2*cos(2*π/11) 1.588774170824 = 2*cos(16*π/77) 1.484468246009 = 2*cos(18*π/77) 1.370283429936 = 2*cos(20*π/77) 1.246979603717 = 2*cos(22*π/77) = 2*cos(2*π/7) 1.115377333974 = 2*cos(24*π/77) 0.976352412097 = 2*cos(26*π/77) 0.830830026004 = 2*cos(28*π/77) = 2*cos(4*π/11) 0.679778603165 = 2*cos(30*π/77) 0.524203365878 = 2*cos(32*π/77) 0.365139641679 = 2*cos(34*π/77) 0.203645973408 = 2*cos(36*π/77) 0.040797074783 = 2*cos(38*π/77) -0.122323321644 = 2*cos(40*π/77) -0.284629676547 = 2*cos(42*π/77) = 2*cos(6*π/11) -0.445041867913 = 2*cos(44*π/77) = 2*cos(4*π/7) -0.602492379074 = 2*cos(46*π/77) -0.755933402858 = 2*cos(48*π/77) -0.904343814565 = 2*cos(50*π/77) -1.046735967393 = 2*cos(52*π/77) -1.182162265057 = 2*cos(54*π/77) -1.309721467891 = 2*cos(56*π/77) = 2*cos(8*π/11) -1.428564690444 = 2*cos(58*π/77) -1.537901050679 = 2*cos(60*π/77) -1.637002933157 = 2*cos(62*π/77) -1.725210831202 = 2*cos(64*π/77) -1.801937735805 = 2*cos(66*π/77) = 2*cos(6*π/7) -1.866673042075 = 2*cos(68*π/77) -1.918985947229 = 2*cos(70*π/77) = 2*cos(10*π/11) -1.958528317515 = 2*cos(72*π/77) -1.985037004982 = 2*cos(74*π/77) -1.998335598689 = 2*cos(76*π/77) 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 C77v point group is 154, and the order of the principal axis (C77) is 77. The group has 40 irreducible representations. β The C77v point group is isomorphic to D77. γ The C77v point group is generated by two symmetry elements, C77 and any σv. Also, the group may be generated from two σv planes (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The lowest nonvanishing multipole moment in C77v is 2 (dipole moment). ε This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. ζ 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.
| C75v | ||
| C76v | ||
| C77 | C77v | C77h D77 D77h D77d |
| C78v | ||
| C79v |
This Character Table for the C77v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.