D17 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' <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 A2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 ..T ..T ..... ......T ......... ..........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 d+g+i+k+m z2, z4, z6 A2 R+p+f+h+j+l Rz, z, z3, z5 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 D17 point group is 34, and the order of the principal axis (C17) is 17. The group has 10 irreducible representations. β The D17 point group is isomorphic to C17v. γ The D17 point group is generated by two symmetry elements, C17 and a perpendicular C2′. Also, the group may be generated from any two C2′ axes. 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. ε The lowest nonvanishing multipole moment in D17 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 The point group is chiral, as it does not contain any mirroring operation. θ 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. ι 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. κ 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.
| D15 | ||
| D16 | ||
| C17 C17v C17h | D17 | D17h D17d |
| D18 | ||
| D19 |
This Character Table for the D17 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.