D20h E 2 C20 2 C10 2 C20^3 2 C5 2 C4 2 C10^3 2 C20^7 2 C5^2 2 C20^9 C2 10 C2' 10 C2" i 2 S20 2 S10 2 S20^3 2 S5 2 S4 2 S10^3 2 S20^7 2 S5^2 2 S20^9 sh 10 sv 10 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1g 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ... ... ....T ....... ........T ........... ............T A2g 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 ..T ... ..... ....... ......... ........... ............. B1g 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 1.0000 -1.0000 ... ... ..... ....... ......... ........... ............. B2g 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 ... ... ..... ....... ......... ........... ............. E1g 2.0000 1.9021 1.6180 1.1755 0.6180 0.0000 -0.6180 -1.1755 -1.6180 -1.9021 -2.0000 0.0000 0.0000 2.0000 -1.9021 -1.6180 -1.1755 -0.6180 0.0000 0.6180 1.1755 1.6180 1.9021 -2.0000 0.0000 0.0000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g 2.0000 1.6180 0.6180 -0.6180 -1.6180 -2.0000 -1.6180 -0.6180 0.6180 1.6180 2.0000 0.0000 0.0000 2.0000 1.6180 0.6180 -0.6180 -1.6180 -2.0000 -1.6180 -0.6180 0.6180 1.6180 2.0000 0.0000 0.0000 ... ... TT... ....... ....TT... ........... ........TT... E3g 2.0000 1.1755 -0.6180 -1.9021 -1.6180 0.0000 1.6180 1.9021 0.6180 -1.1755 -2.0000 0.0000 0.0000 2.0000 -1.1755 0.6180 1.9021 1.6180 0.0000 -1.6180 -1.9021 -0.6180 1.1755 -2.0000 0.0000 0.0000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g 2.0000 0.6180 -1.6180 -1.6180 0.6180 2.0000 0.6180 -1.6180 -1.6180 0.6180 2.0000 0.0000 0.0000 2.0000 0.6180 -1.6180 -1.6180 0.6180 2.0000 0.6180 -1.6180 -1.6180 0.6180 2.0000 0.0000 0.0000 ... ... ..... ....... TT....... ........... ....TT....... E5g 2.0000 0.0000 -2.0000 0.0000 2.0000 0.0000 -2.0000 0.0000 2.0000 0.0000 -2.0000 0.0000 0.0000 2.0000 0.0000 2.0000 0.0000 -2.0000 0.0000 2.0000 0.0000 -2.0000 0.0000 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ..TT......... E6g 2.0000 -0.6180 -1.6180 1.6180 0.6180 -2.0000 0.6180 1.6180 -1.6180 -0.6180 2.0000 0.0000 0.0000 2.0000 -0.6180 -1.6180 1.6180 0.6180 -2.0000 0.6180 1.6180 -1.6180 -0.6180 2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... TT........... E7g 2.0000 -1.1755 -0.6180 1.9021 -1.6180 0.0000 1.6180 -1.9021 0.6180 1.1755 -2.0000 0.0000 0.0000 2.0000 1.1755 0.6180 -1.9021 1.6180 0.0000 -1.6180 1.9021 -0.6180 -1.1755 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E8g 2.0000 -1.6180 0.6180 0.6180 -1.6180 2.0000 -1.6180 0.6180 0.6180 -1.6180 2.0000 0.0000 0.0000 2.0000 -1.6180 0.6180 0.6180 -1.6180 2.0000 -1.6180 0.6180 0.6180 -1.6180 2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E9g 2.0000 -1.9021 1.6180 -1.1755 0.6180 0.0000 -0.6180 1.1755 -1.6180 1.9021 -2.0000 0.0000 0.0000 2.0000 1.9021 -1.6180 1.1755 -0.6180 0.0000 0.6180 -1.1755 1.6180 -1.9021 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. A1u 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 ... ... ..... ....... ......... ........... ............. A2u 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 1.0000 1.0000 ... ..T ..... ......T ......... ..........T ............. B1u 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 1.0000 -1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 -1.0000 1.0000 ... ... ..... ....... ......... ........... ............. B2u 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 ... ... ..... ....... ......... ........... ............. E1u 2.0000 1.9021 1.6180 1.1755 0.6180 0.0000 -0.6180 -1.1755 -1.6180 -1.9021 -2.0000 0.0000 0.0000 -2.0000 1.9021 1.6180 1.1755 0.6180 0.0000 -0.6180 -1.1755 -1.6180 -1.9021 2.0000 0.0000 0.0000 ... TT. ..... ....TT. ......... ........TT. ............. E2u 2.0000 1.6180 0.6180 -0.6180 -1.6180 -2.0000 -1.6180 -0.6180 0.6180 1.6180 2.0000 0.0000 0.0000 -2.0000 -1.6180 -0.6180 0.6180 1.6180 2.0000 1.6180 0.6180 -0.6180 -1.6180 -2.0000 0.0000 0.0000 ... ... ..... ..TT... ......... ......TT... ............. E3u 2.0000 1.1755 -0.6180 -1.9021 -1.6180 0.0000 1.6180 1.9021 0.6180 -1.1755 -2.0000 0.0000 0.0000 -2.0000 1.1755 -0.6180 -1.9021 -1.6180 0.0000 1.6180 1.9021 0.6180 -1.1755 2.0000 0.0000 0.0000 ... ... ..... TT..... ......... ....TT..... ............. E4u 2.0000 0.6180 -1.6180 -1.6180 0.6180 2.0000 0.6180 -1.6180 -1.6180 0.6180 2.0000 0.0000 0.0000 -2.0000 -0.6180 1.6180 1.6180 -0.6180 -2.0000 -0.6180 1.6180 1.6180 -0.6180 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ..TT....... ............. E5u 2.0000 0.0000 -2.0000 0.0000 2.0000 0.0000 -2.0000 0.0000 2.0000 0.0000 -2.0000 0.0000 0.0000 -2.0000 0.0000 -2.0000 0.0000 2.0000 0.0000 -2.0000 0.0000 2.0000 0.0000 2.0000 0.0000 0.0000 ... ... ..... ....... ......... TT......... ............. E6u 2.0000 -0.6180 -1.6180 1.6180 0.6180 -2.0000 0.6180 1.6180 -1.6180 -0.6180 2.0000 0.0000 0.0000 -2.0000 0.6180 1.6180 -1.6180 -0.6180 2.0000 -0.6180 -1.6180 1.6180 0.6180 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E7u 2.0000 -1.1755 -0.6180 1.9021 -1.6180 0.0000 1.6180 -1.9021 0.6180 1.1755 -2.0000 0.0000 0.0000 -2.0000 -1.1755 -0.6180 1.9021 -1.6180 0.0000 1.6180 -1.9021 0.6180 1.1755 2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E8u 2.0000 -1.6180 0.6180 0.6180 -1.6180 2.0000 -1.6180 0.6180 0.6180 -1.6180 2.0000 0.0000 0.0000 -2.0000 1.6180 -0.6180 -0.6180 1.6180 -2.0000 1.6180 -0.6180 -0.6180 1.6180 -2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. E9u 2.0000 -1.9021 1.6180 -1.1755 0.6180 0.0000 -0.6180 1.1755 -1.6180 1.9021 -2.0000 0.0000 0.0000 -2.0000 -1.9021 1.6180 -1.1755 0.6180 0.0000 -0.6180 1.1755 -1.6180 1.9021 2.0000 0.0000 0.0000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.902113032590 = 2*cos(2*π/20) = 2*cos(π/10) = (√10+2*√5)/2 1.618033988750 = 2*cos(4*π/20) = 2*cos(π/5) = (√5+1)/2 1.175570504585 = 2*cos(6*π/20) = 2*cos(3*π/10) = (√10−2*√5)/2 0.618033988750 = 2*cos(8*π/20) = 2*cos(2*π/5) = (√5−1)/2 Symmetry of Rotations and Cartesian products A1g d+g+i+k+m z2, z4, z6 A2g R Rz B1g m B2g m E1g R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} E2g d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E3g g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4g g+i+k+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5g 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)} E6g i+k+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7g k+m E8g k+m E9g m A2u p+f+h+j+l z, z3, z5 E1u p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E2u f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E3u f+h+j+l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E4u h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E5u h+j+l {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)} E6u j+l E7u j+l E8u l E9u l Notes: α The order of the D20h point group is 80, and the order of the principal axis (C20) is 20. The group has 26 irreducible representations. β The D20h point group is generated by three symmetry elements that are canonically chosen C20, C2′ and i. Other choices include σh instead of i, or any of C2″, σv or σd instead of C2′. Also, some ternary combinations of C2′, C2″, σv and σd act as generators. Lastly, the S20 can be chosen, together with i or σh and any one of C2′, C2″, σv or σd. γ 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 C2′ contains both the x and y axes. δ There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation). By convention, the set denoted as σv contains both the xz and the yz planes. ε The lowest nonvanishing multipole moment in D20h 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. Regular polygons of order 10,20,40,80 etc. are easily derived from the regular pentagon by successive halving of angles.
| D18h | ||
| D19h | ||
| C20 C20v C20h D20 | D20h | D20d S20 |
| D21h | ||
| D22h |
This Character Table for the D20h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.