C17v E 2 C17 2 C17^2 2 C17^3 2 C17^4 2 C17^5 2 C17^6 2 C17^7 2 C17^8 17 sv <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ... ..T ....T ......T ........T ..........T ............T A2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 ..T ... ..... ....... ......... ........... ............. E1 2.0000 1.8649 1.4780 0.8914 0.1845 -0.5473 -1.2052 -1.7004 -1.9659 0.0000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.0000 1.4780 0.1845 -1.2052 -1.9659 -1.7004 -0.5473 0.8914 1.8649 0.0000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.0000 0.8914 -1.2052 -1.9659 -0.5473 1.4780 1.8649 0.1845 -1.7004 0.0000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.0000 0.1845 -1.9659 -0.5473 1.8649 0.8914 -1.7004 -1.2052 1.4780 0.0000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.0000 -0.5473 -1.7004 1.4780 0.8914 -1.9659 0.1845 1.8649 -1.2052 0.0000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.0000 -1.2052 -0.5473 1.8649 -1.7004 0.1845 1.4780 -1.9659 0.8914 0.0000 ... ... ..... ....... ......... ........... TT........... E7 2.0000 -1.7004 0.8914 0.1845 -1.2052 1.8649 -1.9659 1.4780 -0.5473 0.0000 ... ... ..... ....... ......... ........... ............. E8 2.0000 -1.9659 1.8649 -1.7004 1.4780 -1.2052 0.8914 -0.5473 0.1845 0.0000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.864944458809 = 2*cos(2*π/17) = (−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 1.478017834441 = 2*cos(4*π/17) = (−1+√17−√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 0.891476711553 = 2*cos(6*π/17) = (−1−√17+√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.184536718927 = 2*cos(8*π/17) = (−1+√17+√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 -0.547325980144 = 2*cos(10*π/17) = −(1+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 -1.205269272759 = 2*cos(12*π/17) = −(1+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 -1.700434271459 = 2*cos(14*π/17) = −(1+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 -1.965946199368 = 2*cos(16*π/17) = −(1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 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+2m E8 k+2l+2m Notes: α The order of the C17v point group is 34, and the order of the principal axis (C17) is 17. The group has 10 irreducible representations. β The C17v point group is isomorphic to D17. γ The C17v point group is generated by two symmetry elements, C17 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 C17v 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. θ The point group corresponds to a constructible polygon, as the order of the principal axis is a product of any number of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers. ι That a regular 17-gon can be constructed with compass and ruler was unknown to mathematicians until Gauss proved it in 1796. The first actual construction was performed thirty years later by Johannes Erchinger in 1825.
| C15v | ||
| C16v | ||
| C17 | C17v | C17h D17 D17h D17d |
| C18v | ||
| C19v |
This Character Table for the C17v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.