D19 E 2 C19 2 C19^2 2 C19^3 2 C19^4 2 C19^5 2 C19^6 2 C19^7 2 C19^8 2 C19^9 19 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 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 -1.0000 ..T ..T ..... ......T ......... ..........T ............. E1 2.0000 1.8916 1.5782 1.0939 0.4909 -0.1651 -0.8033 -1.3545 -1.7589 -1.9727 0.0000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.0000 1.5782 0.4909 -0.8033 -1.7589 -1.9727 -1.3545 -0.1651 1.0939 1.8916 0.0000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.0000 1.0939 -0.8033 -1.9727 -1.3545 0.4909 1.8916 1.5782 -0.1651 -1.7589 0.0000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.0000 0.4909 -1.7589 -1.3545 1.0939 1.8916 -0.1651 -1.9727 -0.8033 1.5782 0.0000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.0000 -0.1651 -1.9727 0.4909 1.8916 -0.8033 -1.7589 1.0939 1.5782 -1.3545 0.0000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.0000 -0.8033 -1.3545 1.8916 -0.1651 -1.7589 1.5782 0.4909 -1.9727 1.0939 0.0000 ... ... ..... ....... ......... ........... TT........... E7 2.0000 -1.3545 -0.1651 1.5782 -1.9727 1.0939 0.4909 -1.7589 1.8916 -0.8033 0.0000 ... ... ..... ....... ......... ........... ............. E8 2.0000 -1.7589 1.0939 -0.1651 -0.8033 1.5782 -1.9727 1.8916 -1.3545 0.4909 0.0000 ... ... ..... ....... ......... ........... ............. E9 2.0000 -1.9727 1.8916 -1.7589 1.5782 -1.3545 1.0939 -0.8033 0.4909 -0.1651 0.0000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.891634483401 = 2*cos(2*π/19) 1.578281018793 = 2*cos(4*π/19) 1.093896316245 = 2*cos(6*π/19) 0.490970974282 = 2*cos(8*π/19) -0.165158690945 = 2*cos(10*π/19) -0.803390849306 = 2*cos(12*π/19) -1.354563143251 = 2*cos(14*π/19) -1.758947502413 = 2*cos(16*π/19) -1.972722606805 = 2*cos(18*π/19) 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+m E8 k+l+m E9 l+2m Notes: α The order of the D19 point group is 38, and the order of the principal axis (C19) is 19. The group has 11 irreducible representations. β The D19 point group is isomorphic to C19v. γ The D19 point group is generated by two symmetry elements, C19 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 D19 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 polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.
| D17 | ||
| D18 | ||
| C19 C19v C19h | D19 | D19h D19d |
| D20 | ||
| D21 |
This Character Table for the D19 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.