D17h 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 C2' sh 2 S17 2 S17^3 2 S17^5 2 S17^7 2 S17^9 2 S17^11 2 S17^13 2 S17^15 17 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 ... ... ....T ....... ........T ........... ............T 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 ... ... ..... ....... ......... ........... ............. 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 ..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 ... ..T ..... ......T ......... ..........T ............. E1' 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 0.00000 2.00000 1.86494 0.89148 -0.54733 -1.70043 -1.96595 -1.20527 0.18454 1.47802 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E1" 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 0.00000 -2.00000 -1.86494 -0.89148 0.54733 1.70043 1.96595 1.20527 -0.18454 -1.47802 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2' 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 0.00000 2.00000 1.47802 -1.20527 -1.70043 0.89148 1.86494 -0.54733 -1.96595 0.18454 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E2" 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 0.00000 -2.00000 -1.47802 1.20527 1.70043 -0.89148 -1.86494 0.54733 1.96595 -0.18454 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E3' 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 0.00000 2.00000 0.89148 -1.96595 1.47802 0.18454 -1.70043 1.86494 -0.54733 -1.20527 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E3" 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 0.00000 -2.00000 -0.89148 1.96595 -1.47802 -0.18454 1.70043 -1.86494 0.54733 1.20527 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4' 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 0.00000 2.00000 0.18454 -0.54733 0.89148 -1.20527 1.47802 -1.70043 1.86494 -1.96595 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E4" 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 0.00000 -2.00000 -0.18454 0.54733 -0.89148 1.20527 -1.47802 1.70043 -1.86494 1.96595 0.00000 ... ... ..... ....... ......... ..TT....... ............. E5' 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 0.00000 2.00000 -0.54733 1.47802 -1.96595 1.86494 -1.20527 0.18454 0.89148 -1.70043 0.00000 ... ... ..... ....... ......... TT......... ............. E5" 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 0.00000 -2.00000 0.54733 -1.47802 1.96595 -1.86494 1.20527 -0.18454 -0.89148 1.70043 0.00000 ... ... ..... ....... ......... ........... ..TT......... E6' 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.00000 2.00000 -1.20527 1.86494 0.18454 -1.96595 0.89148 1.47802 -1.70043 -0.54733 0.00000 ... ... ..... ....... ......... ........... TT........... E6" 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.00000 -2.00000 1.20527 -1.86494 -0.18454 1.96595 -0.89148 -1.47802 1.70043 0.54733 0.00000 ... ... ..... ....... ......... ........... ............. E7' 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 0.00000 2.00000 -1.70043 0.18454 1.86494 1.47802 -0.54733 -1.96595 -1.20527 0.89148 0.00000 ... ... ..... ....... ......... ........... ............. E7" 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 0.00000 -2.00000 1.70043 -0.18454 -1.86494 -1.47802 0.54733 1.96595 1.20527 -0.89148 0.00000 ... ... ..... ....... ......... ........... ............. E8' 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.00000 2.00000 -1.96595 -1.70043 -1.20527 -0.54733 0.18454 0.89148 1.47802 1.86494 0.00000 ... ... ..... ....... ......... ........... ............. E8" 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.00000 -2.00000 1.96595 1.70043 1.20527 0.54733 -0.18454 -0.89148 -1.47802 -1.86494 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.965946199368 = 2*cos(2*π/34) = 2*cos(π/17) = (1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.864944458809 = 2*cos(4*π/34) = 2*cos(2*π/17) = (−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 1.700434271459 = 2*cos(6*π/34) = 2*cos(3*π/17) = (1+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 1.478017834441 = 2*cos(8*π/34) = 2*cos(4*π/17) = (−1+√17−√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.205269272759 = 2*cos(10*π/34) = 2*cos(5*π/17) = (1+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 0.891476711553 = 2*cos(12*π/34) = 2*cos(6*π/17) = (−1−√17+√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.547325980144 = 2*cos(14*π/34) = 2*cos(7*π/17) = (1+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.184536718927 = 2*cos(16*π/34) = 2*cos(8*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' d+g+i+k+m z2, z4, z6 A2' R Rz A2" p+f+h+j+l z, z3, z5 E1' p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E1" R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} E2' d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E2" f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E3' f+h+j+l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E3" g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4' g+i+k+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E4" h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E5' h+j+l {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)} E5" i+k+m {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6' i+k+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E6" j+l E7' j+l+m E7" k+m E8' k+l+m E8" l+m Notes: α The order of the D17h point group is 68, and the order of the principal axis (S17) is 34. The group has 20 irreducible representations. β The D17h point group is isomorphic to D17d, C34v and D34. γ The D17h point group is generated by two symmetry elements, S17 and either a perpendicular C2′ or a vertical σv. Also, the group may be generated from any two σv planes, or any σv and a non-coplanar C2′. The canonical choice, however, is to use redundant generators: C17, C2′ and σh. δ The group contains one set of C2′ symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set. Similarly, the single set of symmetry planes denoted σd contains the xz plane but not the yz plane. ε The lowest nonvanishing multipole moment in D17h is 4 (quadrupole 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.
| D15h | ||
| D16h | ||
| C17 C17v C17h D17 | D17h | D17d |
| D18h | ||
| D19h |
This Character Table for the D17h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.