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E49 2.00000 -1.55692 0.42401 0.89677 -1.82021 1.93717 -1.19581 -0.07538 1.31317 -1.96913 1.75261 -0.75956 -0.57004 1.64707 -1.99432 1.45794 -0.27558 -1.02888 1.87747 -1.89420 1.07165 0.22571 -1.42307 1.98990 -1.67506 0.61803 0.71282 -1.72785 1.97730 -1.35067 0.12558 1.15515 -1.92406 1.84046 -0.94141 -0.37476 1.52489 -1.99937 1.58798 -0.47300 -0.85156 1.79881 -1.94905 1.23572 0.02513 -1.27485 1.95971 -1.77627 0.80581 0.52168 -1.61803 1.99747 -1.49188 0.32527 0.98545 -1.85955 1.90973 -1.11375 -0.17570 1.38731 -1.98423 1.70199 -0.66564 0.00000 ... ... ..... ....... ......... ........... ............. E50 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.00000 ... ... ..... ....... ......... ........... ............. E51 2.00000 -1.67506 0.80581 0.32527 -1.35067 1.93717 -1.89420 1.23572 -0.17570 -0.94141 1.75261 -1.99432 1.58798 -0.66564 -0.47300 1.45794 -1.96913 1.84046 -1.11375 0.02513 1.07165 -1.82021 1.97730 -1.49188 0.52168 0.61803 -1.55692 1.98990 -1.77627 0.98545 0.12558 -1.19581 1.87747 -1.94905 1.38731 -0.37476 -0.75956 1.64707 -1.99937 1.70199 -0.85156 -0.27558 1.31317 -1.92406 1.90973 -1.27485 0.22571 0.89677 -1.72785 1.99747 -1.61803 0.71282 0.42401 -1.42307 1.95971 -1.85955 1.15515 -0.07538 -1.02888 1.79881 -1.98423 1.52489 -0.57004 0.00000 ... ... ..... ....... ......... ........... ............. E52 2.00000 -1.72785 0.98545 0.02513 -1.02888 1.75261 -1.99937 1.70199 -0.94141 -0.07538 1.07165 -1.77627 1.99747 -1.67506 0.89677 0.12558 -1.11375 1.79881 -1.99432 1.64707 -0.85156 -0.17570 1.15515 -1.82021 1.98990 -1.61803 0.80581 0.22571 -1.19581 1.84046 -1.98423 1.58798 -0.75956 -0.27558 1.23572 -1.85955 1.97730 -1.55692 0.71282 0.32527 -1.27485 1.87747 -1.96913 1.52489 -0.66564 -0.37476 1.31317 -1.89420 1.95971 -1.49188 0.61803 0.42401 -1.35067 1.90973 -1.94905 1.45794 -0.57004 -0.47300 1.38731 -1.92406 1.93717 -1.42307 0.52168 0.00000 ... ... ..... ....... ......... ........... ............. E53 2.00000 -1.77627 1.15515 -0.27558 -0.66564 1.45794 -1.92406 1.95971 -1.55692 0.80581 0.12558 -1.02888 1.70199 -1.99432 1.84046 -1.27485 0.42401 0.52168 -1.35067 1.87747 -1.98423 1.64707 -0.94141 0.02513 0.89677 -1.61803 1.97730 -1.89420 1.38731 -0.57004 -0.37476 1.23572 -1.82021 1.99747 -1.72785 1.07165 -0.17570 -0.75956 1.52489 -1.94905 1.93717 -1.49188 0.71282 0.22571 -1.11375 1.75261 -1.99937 1.79881 -1.19581 0.32527 0.61803 -1.42307 1.90973 -1.96913 1.58798 -0.85156 -0.07538 0.98545 -1.67506 1.98990 -1.85955 1.31317 -0.47300 0.00000 ... ... ..... ....... ......... ........... ............. E54 2.00000 -1.82021 1.31317 -0.57004 -0.27558 1.07165 -1.67506 1.97730 -1.92406 1.52489 -0.85156 0.02513 0.80581 -1.49188 1.90973 -1.98423 1.70199 -1.11375 0.32527 0.52168 -1.27485 1.79881 -1.99937 1.84046 -1.35067 0.61803 0.22571 -1.02888 1.64707 -1.96913 1.93717 -1.55692 0.89677 -0.07538 -0.75956 1.45794 -1.89420 1.98990 -1.72785 1.15515 -0.37476 -0.47300 1.23572 -1.77627 1.99747 -1.85955 1.38731 -0.66564 -0.17570 0.98545 -1.61803 1.95971 -1.94905 1.58798 -0.94141 0.12558 0.71282 -1.42307 1.87747 -1.99432 1.75261 -1.19581 0.42401 0.00000 ... ... ..... ....... ......... ........... ............. E55 2.00000 -1.85955 1.45794 -0.85156 0.12558 0.61803 -1.27485 1.75261 -1.98423 1.93717 -1.61803 1.07165 -0.37476 -0.37476 1.07165 -1.61803 1.93717 -1.98423 1.75261 -1.27485 0.61803 0.12558 -0.85156 1.45794 -1.85955 2.00000 -1.85955 1.45794 -0.85156 0.12558 0.61803 -1.27485 1.75261 -1.98423 1.93717 -1.61803 1.07165 -0.37476 -0.37476 1.07165 -1.61803 1.93717 -1.98423 1.75261 -1.27485 0.61803 0.12558 -0.85156 1.45794 -1.85955 2.00000 -1.85955 1.45794 -0.85156 0.12558 0.61803 -1.27485 1.75261 -1.98423 1.93717 -1.61803 1.07165 -0.37476 0.00000 ... ... ..... ....... ......... ........... ............. E56 2.00000 -1.89420 1.58798 -1.11375 0.52168 0.12558 -0.75956 1.31317 -1.72785 1.95971 -1.98423 1.79881 -1.42307 0.89677 -0.27558 -0.37476 0.98545 -1.49188 1.84046 -1.99432 1.93717 -1.67506 1.23572 -0.66564 0.02513 0.61803 -1.19581 1.64707 -1.92406 1.99747 -1.85955 1.52489 -1.02888 0.42401 0.22571 -0.85156 1.38731 -1.77627 1.97730 -1.96913 1.75261 -1.35067 0.80581 -0.17570 -0.47300 1.07165 -1.55692 1.87747 -1.99937 1.90973 -1.61803 1.15515 -0.57004 -0.07538 0.71282 -1.27485 1.70199 -1.94905 1.98990 -1.82021 1.45794 -0.94141 0.32527 0.00000 ... ... ..... ....... ......... ........... ............. E57 2.00000 -1.92406 1.70199 -1.35067 0.89677 -0.37476 -0.17570 0.71282 -1.19581 1.58798 -1.85955 1.98990 -1.96913 1.79881 -1.49188 1.07165 -0.57004 0.02513 0.52168 -1.02888 1.45794 -1.77627 1.95971 -1.99432 1.87747 -1.61803 1.23572 -0.75956 0.22571 0.32527 -0.85156 1.31317 -1.67506 1.90973 -1.99937 1.93717 -1.72785 1.38731 -0.94141 0.42401 0.12558 -0.66564 1.15515 -1.55692 1.84046 -1.98423 1.97730 -1.82021 1.52489 -1.11375 0.61803 -0.07538 -0.47300 0.98545 -1.42307 1.75261 -1.94905 1.99747 -1.89420 1.64707 -1.27485 0.80581 -0.27558 0.00000 ... ... ..... ....... ......... ........... ............. E58 2.00000 -1.94905 1.79881 -1.55692 1.23572 -0.85156 0.42401 0.02513 -0.47300 0.89677 -1.27485 1.58798 -1.82021 1.95971 -1.99937 1.93717 -1.77627 1.52489 -1.19581 0.80581 -0.37476 -0.07538 0.52168 -0.94141 1.31317 -1.61803 1.84046 -1.96913 1.99747 -1.92406 1.75261 -1.49188 1.15515 -0.75956 0.32527 0.12558 -0.57004 0.98545 -1.35067 1.64707 -1.85955 1.97730 -1.99432 1.90973 -1.72785 1.45794 -1.11375 0.71282 -0.27558 -0.17570 0.61803 -1.02888 1.38731 -1.67506 1.87747 -1.98423 1.98990 -1.89420 1.70199 -1.42307 1.07165 -0.66564 0.22571 0.00000 ... ... ..... ....... ......... ........... ............. E59 2.00000 -1.96913 1.87747 -1.72785 1.52489 -1.27485 0.98545 -0.66564 0.32527 0.02513 -0.37476 0.71282 -1.02888 1.31317 -1.55692 1.75261 -1.89420 1.97730 -1.99937 1.95971 -1.85955 1.70199 -1.49188 1.23572 -0.94141 0.61803 -0.27558 -0.07538 0.42401 -0.75956 1.07165 -1.35067 1.58798 -1.77627 1.90973 -1.98423 1.99747 -1.94905 1.84046 -1.67506 1.45794 -1.19581 0.89677 -0.57004 0.22571 0.12558 -0.47300 0.80581 -1.11375 1.38731 -1.61803 1.79881 -1.92406 1.98990 -1.99432 1.93717 -1.82021 1.64707 -1.42307 1.15515 -0.85156 0.52168 -0.17570 0.00000 ... ... ..... ....... ......... ........... ............. E60 2.00000 -1.98423 1.93717 -1.85955 1.75261 -1.61803 1.45794 -1.27485 1.07165 -0.85156 0.61803 -0.37476 0.12558 0.12558 -0.37476 0.61803 -0.85156 1.07165 -1.27485 1.45794 -1.61803 1.75261 -1.85955 1.93717 -1.98423 2.00000 -1.98423 1.93717 -1.85955 1.75261 -1.61803 1.45794 -1.27485 1.07165 -0.85156 0.61803 -0.37476 0.12558 0.12558 -0.37476 0.61803 -0.85156 1.07165 -1.27485 1.45794 -1.61803 1.75261 -1.85955 1.93717 -1.98423 2.00000 -1.98423 1.93717 -1.85955 1.75261 -1.61803 1.45794 -1.27485 1.07165 -0.85156 0.61803 -0.37476 0.12558 0.00000 ... ... ..... ....... ......... ........... ............. E61 2.00000 -1.99432 1.97730 -1.94905 1.90973 -1.85955 1.79881 -1.72785 1.64707 -1.55692 1.45794 -1.35067 1.23572 -1.11375 0.98545 -0.85156 0.71282 -0.57004 0.42401 -0.27558 0.12558 0.02513 -0.17570 0.32527 -0.47300 0.61803 -0.75956 0.89677 -1.02888 1.15515 -1.27485 1.38731 -1.49188 1.58798 -1.67506 1.75261 -1.82021 1.87747 -1.92406 1.95971 -1.98423 1.99747 -1.99937 1.98990 -1.96913 1.93717 -1.89420 1.84046 -1.77627 1.70199 -1.61803 1.52489 -1.42307 1.31317 -1.19581 1.07165 -0.94141 0.80581 -0.66564 0.52168 -0.37476 0.22571 -0.07538 0.00000 ... ... ..... ....... ......... ........... ............. E62 2.00000 -1.99937 1.99747 -1.99432 1.98990 -1.98423 1.97730 -1.96913 1.95971 -1.94905 1.93717 -1.92406 1.90973 -1.89420 1.87747 -1.85955 1.84046 -1.82021 1.79881 -1.77627 1.75261 -1.72785 1.70199 -1.67506 1.64707 -1.61803 1.58798 -1.55692 1.52489 -1.49188 1.45794 -1.42307 1.38731 -1.35067 1.31317 -1.27485 1.23572 -1.19581 1.15515 -1.11375 1.07165 -1.02888 0.98545 -0.94141 0.89677 -0.85156 0.80581 -0.75956 0.71282 -0.66564 0.61803 -0.57004 0.52168 -0.47300 0.42401 -0.37476 0.32527 -0.27558 0.22571 -0.17570 0.12558 -0.07538 0.02513 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.997473913212 = 2*cos(2*π/125) 1.989902033963 = 2*cos(4*π/125) 1.977303489476 = 2*cos(6*π/125) 1.959710104768 = 2*cos(8*π/125) 1.937166322257 = 2*cos(10*π/125) = 2*cos(2*π/25) 1.909729089493 = 2*cos(12*π/125) 1.877467715308 = 2*cos(14*π/125) 1.840463694732 = 2*cos(16*π/125) 1.798810503133 = 2*cos(18*π/125) 1.752613360088 = 2*cos(20*π/125) = 2*cos(4*π/25) 1.701988963589 = 2*cos(22*π/125) 1.647065195257 = 2*cos(24*π/125) 1.587980797296 = 2*cos(26*π/125) 1.524885022023 = 2*cos(28*π/125) 1.457937254843 = 2*cos(30*π/125) = 2*cos(6*π/25) 1.387306611626 = 2*cos(32*π/125) 1.313171511506 = 2*cos(34*π/125) 1.235719226181 = 2*cos(36*π/125) 1.155145406845 = 2*cos(38*π/125) 1.071653589958 = 2*cos(40*π/125) = 2*cos(8*π/25) 0.985454683097 = 2*cos(42*π/125) 0.896766432180 = 2*cos(44*π/125) 0.805812871427 = 2*cos(46*π/125) 0.712823757427 = 2*cos(48*π/125) 0.618033988750 = 2*cos(50*π/125) = 2*cos(2*π/5) = (√5−1)/2 0.521683012580 = 2*cos(52*π/125) 0.424014219844 = 2*cos(54*π/125) 0.325274330390 = 2*cos(56*π/125) 0.225712769747 = 2*cos(58*π/125) 0.125581039059 = 2*cos(60*π/125) = 2*cos(12*π/25) 0.025132079767 = 2*cos(62*π/125) -0.075380365340 = 2*cos(64*π/125) -0.175702393101 = 2*cos(66*π/125) -0.275580581369 = 2*cos(68*π/125) -0.374762629171 = 2*cos(70*π/125) = 2*cos(14*π/25) -0.472997994047 = 2*cos(72*π/125) -0.570038524940 = 2*cos(74*π/125) -0.665639089046 = 2*cos(76*π/125) -0.759558191044 = 2*cos(78*π/125) -0.851558583130 = 2*cos(80*π/125) = 2*cos(16*π/25) -0.941407864331 = 2*cos(82*π/125) -1.028879067563 = 2*cos(84*π/125) -1.113751232976 = 2*cos(86*π/125) -1.195809966115 = 2*cos(88*π/125) -1.274847979497 = 2*cos(90*π/125) = 2*cos(18*π/25) -1.350665616242 = 2*cos(92*π/125) -1.423071354419 = 2*cos(94*π/125) -1.491882290848 = 2*cos(96*π/125) -1.556924603134 = 2*cos(98*π/125) -1.618033988750 = 2*cos(100*π/125) = 2*cos(4*π/5) = −(√5+1)/2 -1.675056080084 = 2*cos(102*π/125) -1.727846834386 = 2*cos(104*π/125) -1.776272897627 = 2*cos(106*π/125) -1.820211941370 = 2*cos(108*π/125) -1.859552971777 = 2*cos(110*π/125) = 2*cos(22*π/25) -1.894196609989 = 2*cos(112*π/125) -1.924055343172 = 2*cos(114*π/125) -1.949053745573 = 2*cos(116*π/125) -1.969128669058 = 2*cos(118*π/125) -1.984229402629 = 2*cos(120*π/125) = 2*cos(24*π/25) -1.994317800521 = 2*cos(122*π/125) -1.999368378567 = 2*cos(124*π/125) 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 C125v point group is 250, and the order of the principal axis (C125) is 125. The group has 64 irreducible representations. β The C125v point group is isomorphic to D125. γ The C125v point group is generated by two symmetry elements, C125 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 C125v 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, some of the irrational characters cannot be expressed by a closed algebraic form using real numbers only.
C123v | ||
C124v | ||
C125 | C125v | C125h D125 D125h D125d |
C126v | ||
C127v |
This Character Table for the C125v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.