D15 E 2 C15 2 C15^2 2 C5 2 C15^4 2 C3 2 C5^2 2 C15^7 15 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 ... ... ....T ....... ........T ........... ............T A2 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 1.8270 1.3382 0.6180 -0.2090 -1.0000 -1.6180 -1.9563 0.0000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.0000 1.3382 -0.2090 -1.6180 -1.9563 -1.0000 0.6180 1.8270 0.0000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.0000 0.6180 -1.6180 -1.6180 0.6180 2.0000 0.6180 -1.6180 0.0000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.0000 -0.2090 -1.9563 0.6180 1.8270 -1.0000 -1.6180 1.3382 0.0000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.0000 -1.0000 -1.0000 2.0000 -1.0000 -1.0000 2.0000 -1.0000 0.0000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.0000 -1.6180 0.6180 0.6180 -1.6180 2.0000 -1.6180 0.6180 0.0000 ... ... ..... ....... ......... ........... TT........... E7 2.0000 -1.9563 1.8270 -1.6180 1.3382 -1.0000 0.6180 -0.2090 0.0000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.827090915285 = 2*cos(2*π/15) = (1+√5+√30−6*√5)/4 1.338261212718 = 2*cos(4*π/15) = (1−√5+√30+6*√5)/4 0.618033988750 = 2*cos(6*π/15) = 2*cos(2*π/5) = (√5−1)/2 -0.209056926535 = 2*cos(8*π/15) = −(√30−6*√5−√5−1)/4 -1.618033988750 = 2*cos(12*π/15) = 2*cos(4*π/5) = −(√5+1)/2 -1.956295201468 = 2*cos(14*π/15) = −(√5−1+√30+6*√5)/4 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} 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+m {(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+l+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+k+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})} E7 j+2k+2l+2m Notes: α The order of the D_{15} point group is 30, and the order of the principal axis (C_{15}) is 15. The group has 9 irreducible representations. β The D_{15} point group is isomorphic to C_{15v}. γ The D_{15} point group is generated by two symmetry elements, C_{15} and a perpendicular C_{2}^{′}. Also, the group may be generated from two C_{2}^{′} axes (some pairs will yield smaller groups, though; choosing a minimum angle is safe). 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. ε The lowest nonvanishing multipole moment in D_{15} 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 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 The fact that the regular pentagon is constructible is known since antiquity; Eukleides already discovered a construction for it. The double cosine of 2π/5 is equal to the reciprocal of the Golden Ratio of (1+√5)/2 = 1.61803. The regular 15-gon can be derived from the regular pentagon by overlaying it with a equilateral triangle.
D_{13} | ||
D_{14} | ||
C_{15} C_{15v} C_{15h} | D_{15} | D_{15h} D_{15d} |
D_{16} | ||
D_{17} |
This Character Table for the D_{15} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.