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 z^{2}, z^{4}, z^{6} A1" k+m A2' R+j+l R_{z} A2" p+f+h+j+k+l+m z, z^{3}, z^{5} E1' p+f+h+i+j+2k+l+2m {x, y}, {xz^{2}, yz^{2}}, {xz^{4}, yz^{4}}, {x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, xy(x^{2}−3y^{2})(3x^{2}−y^{2})} E1" R+d+g+i+j+k+2l+m {R_{x}, R_{y}}, {xz, yz}, {xz^{3}, yz^{3}}, {xz^{5}, yz^{5}} E2' d+g+h+i+j+k+2l+m {x^{2}−y^{2}, xy}, {z^{2}(x^{2}−y^{2}), xyz^{2}}, {x(x^{2}−(5+2√5)y^{2})(x^{2}−(5−2√5)y^{2}), y((5+2√5)x^{2}−y^{2})((5−2√5)x^{2}−y^{2})}, {z^{4}(x^{2}−y^{2}), xyz^{4}} E2" f+h+i+j+k+l+2m {z(x^{2}−y^{2}), xyz}, {z^{3}(x^{2}−y^{2}), xyz^{3}}, {xz(x^{2}−(5+2√5)y^{2})(x^{2}−(5−2√5)y^{2}), yz((5+2√5)x^{2}−y^{2})((5−2√5)x^{2}−y^{2})} E3' f+g+h+i+j+k+l+2m {x(x^{2}−3y^{2}), y(3x^{2}−y^{2})}, {(x^{2}−y^{2})^{2}−4x^{2}y^{2}, xy(x^{2}−y^{2})}, {xz^{2}(x^{2}−3y^{2}), yz^{2}(3x^{2}−y^{2})}, {z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz^{2}(x^{2}−y^{2})} E3" g+h+i+j+k+l+m {xz(x^{2}−3y^{2}), yz(3x^{2}−y^{2})}, {z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz(x^{2}−y^{2})}, {xz^{3}(x^{2}−3y^{2}), yz^{3}(3x^{2}−y^{2})} Notes: α The order of the D_{7h} point group is 28, and the order of the principal axis (S_{7}) is 14. The group has 10 irreducible representations. β The D_{7h} point group is isomorphic to D_{7d}, C_{14v} and D_{14}. γ The D_{7h} point group is generated by two symmetry elements, S_{7} and either a perpendicular C_{2}^{′} or a vertical σ_{v}. Also, the group may be generated from any two σ_{v} planes, or any σ_{v} and a non-coplanar C_{2}^{′}. The canonical choice, however, is to use redundant generators: C_{7}, C_{2}^{′} and σ_{h}. δ The group contains one set of C_{2}^{′} 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 σ_{d} contains the xz plane but not the yz plane. ε The lowest nonvanishing multipole moment in D_{7h} 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.
D_{5h} | ||
D_{6h} | ||
C_{7} C_{7v} C_{7h} D_{7} | D_{7h} | D_{7d} |
D_{8h} | ||
D_{9h} |
This Character Table for the D_{7h} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.