D14 E 2 C14 2 C7 2 C14^3 2 C7^2 2 C14^5 2 C7^3 C2 7 C2' 7 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 ............. B1 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 ... ... ..... ....... ......... ........... ............. B2 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 -1.0000 1.0000 ... ... ..... ....... ......... ........... ............. E1 2.0000 1.8019 1.2469 0.4450 -0.4450 -1.2469 -1.8019 -2.0000 0.0000 0.0000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.0000 1.2469 -0.4450 -1.8019 -1.8019 -0.4450 1.2469 2.0000 0.0000 0.0000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.0000 0.4450 -1.8019 -1.2469 1.2469 1.8019 -0.4450 -2.0000 0.0000 0.0000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.0000 -0.4450 -1.8019 1.2469 1.2469 -1.8019 -0.4450 2.0000 0.0000 0.0000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.0000 -1.2469 -0.4450 1.8019 -1.8019 0.4450 1.2469 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.0000 -1.8019 1.2469 -0.4450 -0.4450 1.2469 -1.8019 2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... 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+k+m z^{2}, z^{4}, z^{6} A2 R+p+f+h+j+l R_{z}, z, z^{3}, z^{5} B1 j+k+l+m B2 j+k+l+m E1 R+p+d+f+g+h+i+j+k+l+m {R_{x}, R_{y}}, {x, y}, {xz, yz}, {xz^{2}, yz^{2}}, {xz^{3}, yz^{3}}, {xz^{4}, yz^{4}}, {xz^{5}, yz^{5}} E2 d+f+g+h+i+j+k+l+m {x^{2}−y^{2}, xy}, {z(x^{2}−y^{2}), xyz}, {z^{2}(x^{2}−y^{2}), xyz^{2}}, {z^{3}(x^{2}−y^{2}), xyz^{3}}, {z^{4}(x^{2}−y^{2}), xyz^{4}} E3 f+g+h+i+j+k+l+m {x(x^{2}−3y^{2}), y(3x^{2}−y^{2})}, {xz(x^{2}−3y^{2}), yz(3x^{2}−y^{2})}, {xz^{2}(x^{2}−3y^{2}), yz^{2}(3x^{2}−y^{2})}, {xz^{3}(x^{2}−3y^{2}), yz^{3}(3x^{2}−y^{2})} E4 g+h+i+j+k+l+2m {(x^{2}−y^{2})^{2}−4x^{2}y^{2}, xy(x^{2}−y^{2})}, {z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz(x^{2}−y^{2})}, {z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz^{2}(x^{2}−y^{2})} E5 h+i+j+k+2l+2m {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})}, {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})} E6 i+j+2k+2l+2m {x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, xy(x^{2}−3y^{2})(3x^{2}−y^{2})} Notes: α The order of the D_{14} point group is 28, and the order of the principal axis (C_{14}) is 14. The group has 10 irreducible representations. β The D_{14} point group is isomorphic to D_{7d}, D_{7h} and C_{14v}. γ The D_{14} point group is generated by two symmetry elements, C_{14} and a perpendicular C_{2}^{′} (or, non-canonically, C_{2}^{″}). Also, the group may be generated from a C_{2}^{′} plus a C_{2}^{″} (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation). By convention, the set denoted as C_{2}^{′} has the x axis as a member, while the y axis is a member of the C_{2}^{″} set. ε The lowest nonvanishing multipole moment in D_{14} 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 much 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. κ 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. Therefore, regular polygons of order 14,21,28,35,42 etc. are also inconstructible, and their cosines have no representation in real radicals.
D_{12} | ||
D_{13} | ||
C_{14} C_{14v} C_{14h} | D_{14} | D_{14h} D_{14d} S_{14} |
D_{15} | ||
D_{16} |
This Character Table for the D_{14} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.