D15h E 2 C15 2 C15^2 2 C5 2 C15^4 2 C3 2 C5^2 2 C15^7 15 C2' sh 2 S15 2 S5 2 S3 2 S15^7 2 S5^3 2 S15^11 2 S15^13 15 sv <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1' 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ... ... ....T ....... ........T ........... ............T A1" 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. A2' 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 ..T ... ..... ....... ......... ........... ............. A2" 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 1.00000 ... ..T ..... ......T ......... ..........T ............. E1' 2.00000 1.82709 1.33826 0.61803 -0.20906 -1.00000 -1.61803 -1.95630 0.00000 2.00000 1.82709 0.61803 -1.00000 -1.95630 -1.61803 -0.20906 1.33826 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E1" 2.00000 1.82709 1.33826 0.61803 -0.20906 -1.00000 -1.61803 -1.95630 0.00000 -2.00000 -1.82709 -0.61803 1.00000 1.95630 1.61803 0.20906 -1.33826 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2' 2.00000 1.33826 -0.20906 -1.61803 -1.95630 -1.00000 0.61803 1.82709 0.00000 2.00000 1.33826 -1.61803 -1.00000 1.82709 0.61803 -1.95630 -0.20906 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E2" 2.00000 1.33826 -0.20906 -1.61803 -1.95630 -1.00000 0.61803 1.82709 0.00000 -2.00000 -1.33826 1.61803 1.00000 -1.82709 -0.61803 1.95630 0.20906 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E3' 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 0.00000 2.00000 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E3" 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 0.00000 -2.00000 -0.61803 1.61803 -2.00000 1.61803 -0.61803 -0.61803 1.61803 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4' 2.00000 -0.20906 -1.95630 0.61803 1.82709 -1.00000 -1.61803 1.33826 0.00000 2.00000 -0.20906 0.61803 -1.00000 1.33826 -1.61803 1.82709 -1.95630 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E4" 2.00000 -0.20906 -1.95630 0.61803 1.82709 -1.00000 -1.61803 1.33826 0.00000 -2.00000 0.20906 -0.61803 1.00000 -1.33826 1.61803 -1.82709 1.95630 0.00000 ... ... ..... ....... ......... ..TT....... ............. E5' 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 0.00000 2.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E5" 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 0.00000 -2.00000 1.00000 -2.00000 1.00000 1.00000 -2.00000 1.00000 1.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E6' 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.00000 2.00000 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 0.00000 ... ... ..... ....... ......... ........... TT........... E6" 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.00000 -2.00000 1.61803 -0.61803 -2.00000 -0.61803 1.61803 1.61803 -0.61803 0.00000 ... ... ..... ....... ......... ........... ............. E7' 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 0.00000 2.00000 -1.95630 -1.61803 -1.00000 -0.20906 0.61803 1.33826 1.82709 0.00000 ... ... ..... ....... ......... ........... ............. E7" 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 0.00000 -2.00000 1.95630 1.61803 1.00000 0.20906 -0.61803 -1.33826 -1.82709 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.956295201468 = 2*cos(2*π/30) = 2*cos(π/15) = (√5−1+√30+6*√5)/4 1.827090915285 = 2*cos(4*π/30) = 2*cos(2*π/15) = (1+√5+√30−6*√5)/4 1.618033988750 = 2*cos(6*π/30) = 2*cos(π/5) = (√5+1)/2 1.338261212718 = 2*cos(8*π/30) = 2*cos(4*π/15) = (1−√5+√30+6*√5)/4 0.618033988750 = 2*cos(12*π/30) = 2*cos(2*π/5) = (√5−1)/2 0.209056926535 = 2*cos(14*π/30) = 2*cos(7*π/15) = (√30−6*√5−√5−1)/4 Symmetry of Rotations and Cartesian products A1' d+g+i+k+m z2, z4, z6 A2' R Rz A2" p+f+h+j+l z, z3, z5 E1' p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E1" R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} E2' d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E2" f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E3' f+h+j+l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E3" g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4' g+i+k+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E4" h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E5' h+j+l+m {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)} E5" i+k+m {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6' i+k+l+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E6" j+l+m E7' j+k+l+m E7" k+l+m Notes: α The order of the D15h point group is 60, and the order of the principal axis (S15) is 30. The group has 18 irreducible representations. β The D15h point group is isomorphic to D15d, C30v and D30. γ The D15h point group is generated by two symmetry elements, S15 and either a perpendicular C2′ or a vertical σv. Also, the group may be generated from two σv planes, or a σv and a C2′ (some pairs will yield smaller groups, though; choosing a minimum angle is safe). The canonical choice, however, is to use redundant generators: C15, 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 D15h 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 much less than half the order of the principal axis. θ 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. ι 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.
| D13h | ||
| D14h | ||
| C15 C15v C15h D15 | D15h | D15d |
| D16h | ||
| D17h |
This Character Table for the D15h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.