D7h E 2 C7 2 C7^2 2 C7^3 7 C2' sh 2 S7 2 S7^3 2 S7^5 7 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 A1" 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 ... ... ..... ....... ......... ........... ............. A2' 1.0000 1.0000 1.0000 1.0000 -1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 ..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 ............. E1' 2.0000 1.2469 -0.4450 -1.8019 0.0000 2.0000 1.2469 -1.8019 -0.4450 0.0000 ... TT. ..... ....TT. ......... ........TT. TT........... E1" 2.0000 1.2469 -0.4450 -1.8019 0.0000 -2.0000 -1.2469 1.8019 0.4450 0.0000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2' 2.0000 -0.4450 -1.8019 1.2469 0.0000 2.0000 -0.4450 1.2469 -1.8019 0.0000 ... ... TT... ....... ....TT... TT......... ........TT... E2" 2.0000 -0.4450 -1.8019 1.2469 0.0000 -2.0000 0.4450 -1.2469 1.8019 0.0000 ... ... ..... ..TT... ......... ......TT... ..TT......... E3' 2.0000 -1.8019 1.2469 -0.4450 0.0000 2.0000 -1.8019 -0.4450 1.2469 0.0000 ... ... ..... TT..... TT....... ....TT..... ....TT....... E3" 2.0000 -1.8019 1.2469 -0.4450 0.0000 -2.0000 1.8019 0.4450 -1.2469 0.0000 ... ... ..... ....... ..TT..... ..TT....... ......TT..... Irrational character values: 1.801937735805 = 2*cos(2*π/14) = 2*cos(π/7) 1.246979603717 = 2*cos(4*π/14) = 2*cos(2*π/7) 0.445041867913 = 2*cos(6*π/14) = 2*cos(3*π/7) Symmetry of Rotations and Cartesian products A1' d+g+i+j+k+l+m z2, z4, z6 A1" k+m A2' R+j+l Rz A2" p+f+h+j+k+l+m z, z3, z5 E1' p+f+h+i+j+2k+l+2m {x, y}, {xz2, yz2}, {xz4, yz4}, {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E1" R+d+g+i+j+k+2l+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} E2' d+g+h+i+j+k+2l+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {z4(x2−y2), xyz4} E2" f+h+i+j+k+l+2m {z(x2−y2), xyz}, {z3(x2−y2), xyz3}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E3' f+g+h+i+j+k+l+2m {x(x2−3y2), y(3x2−y2)}, {(x2−y2)2−4x2y2, xy(x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E3" g+h+i+j+k+l+m {xz(x2−3y2), yz(3x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} Notes: α The order of the D7h point group is 28, and the order of the principal axis (S7) is 14. The group has 10 irreducible representations. β The D7h point group is isomorphic to D7d, C14v and D14. γ The D7h point group is generated by two symmetry elements, S7 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: C7, 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 D7h 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 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. ι The regular heptagon is the lowest regular polygon that cannot be constructed by compass and ruler alone, which has mystified mathematicians since antiquity. The reason for the inconstructibility of the heptagon is that cos(2π/7) has an algebraic degree of 3 and thus can not be represented by square roots and integer numbers. An algebraic representation becomes possible if cubic roots and complex numbers are used: 2*cos(2π/7) = (3√28+i*84*√3 + 3√28−i*84*√3 − 2)/6. This complex expression for a real value is hardly useful.
| D5h | ||
| D6h | ||
| C7 C7v C7h D7 | D7h | D7d |
| D8h | ||
| D9h |
This Character Table for the D7h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.