D26h E 2 C26 2 C13 2 C26^3 2 C13^2 2 C26^5 2 C13^3 2 C26^7 2 C13^4 2 C26^9 2 C13^5 2 C26^11 2 C13^6 C2 13 C2' 13 C2" i 2 S26 2 S13 2 S26^3 2 S13^2 2 S26^5 2 S13^3 2 S26^7 2 S13^4 2 S26^9 2 S13^5 2 S26^11 2 S13^6 sh 13 sv 13 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1g 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 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 A2g 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 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 ... ..... ....... ......... ........... ............. B1g 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 -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 ... ... ..... ....... ......... ........... ............. B2g 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 -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 ... ... ..... ....... ......... ........... ............. E1g 2.00000 1.94188 1.77091 1.49702 1.13613 0.70921 0.24107 -0.24107 -0.70921 -1.13613 -1.49702 -1.77091 -1.94188 -2.00000 0.00000 0.00000 2.00000 -1.94188 -1.77091 -1.49702 -1.13613 -0.70921 -0.24107 0.24107 0.70921 1.13613 1.49702 1.77091 1.94188 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 0.00000 0.00000 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3g 2.00000 1.49702 0.24107 -1.13613 -1.94188 -1.77091 -0.70921 0.70921 1.77091 1.94188 1.13613 -0.24107 -1.49702 -2.00000 0.00000 0.00000 2.00000 -1.49702 -0.24107 1.13613 1.94188 1.77091 0.70921 -0.70921 -1.77091 -1.94188 -1.13613 0.24107 1.49702 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 0.00000 0.00000 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5g 2.00000 0.70921 -1.49702 -1.77091 0.24107 1.94188 1.13613 -1.13613 -1.94188 -0.24107 1.77091 1.49702 -0.70921 -2.00000 0.00000 0.00000 2.00000 -0.70921 1.49702 1.77091 -0.24107 -1.94188 -1.13613 1.13613 1.94188 0.24107 -1.77091 -1.49702 0.70921 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E6g 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 0.00000 0.00000 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7g 2.00000 -0.24107 -1.94188 0.70921 1.77091 -1.13613 -1.49702 1.49702 1.13613 -1.77091 -0.70921 1.94188 0.24107 -2.00000 0.00000 0.00000 2.00000 0.24107 1.94188 -0.70921 -1.77091 1.13613 1.49702 -1.49702 -1.13613 1.77091 0.70921 -1.94188 -0.24107 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8g 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 0.00000 0.00000 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9g 2.00000 -1.13613 -0.70921 1.94188 -1.49702 -0.24107 1.77091 -1.77091 0.24107 1.49702 -1.94188 0.70921 1.13613 -2.00000 0.00000 0.00000 2.00000 1.13613 0.70921 -1.94188 1.49702 0.24107 -1.77091 1.77091 -0.24107 -1.49702 1.94188 -0.70921 -1.13613 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10g 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 0.00000 0.00000 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11g 2.00000 -1.77091 1.13613 -0.24107 -0.70921 1.49702 -1.94188 1.94188 -1.49702 0.70921 0.24107 -1.13613 1.77091 -2.00000 0.00000 0.00000 2.00000 1.77091 -1.13613 0.24107 0.70921 -1.49702 1.94188 -1.94188 1.49702 -0.70921 -0.24107 1.13613 -1.77091 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12g 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 0.00000 0.00000 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. A1u 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 -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 ... ... ..... ....... ......... ........... ............. A2u 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 -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 ............. B1u 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 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 ... ... ..... ....... ......... ........... ............. B2u 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 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 ... ... ..... ....... ......... ........... ............. E1u 2.00000 1.94188 1.77091 1.49702 1.13613 0.70921 0.24107 -0.24107 -0.70921 -1.13613 -1.49702 -1.77091 -1.94188 -2.00000 0.00000 0.00000 -2.00000 1.94188 1.77091 1.49702 1.13613 0.70921 0.24107 -0.24107 -0.70921 -1.13613 -1.49702 -1.77091 -1.94188 2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2u 2.00000 1.77091 1.13613 0.24107 -0.70921 -1.49702 -1.94188 -1.94188 -1.49702 -0.70921 0.24107 1.13613 1.77091 2.00000 0.00000 0.00000 -2.00000 -1.77091 -1.13613 -0.24107 0.70921 1.49702 1.94188 1.94188 1.49702 0.70921 -0.24107 -1.13613 -1.77091 -2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E3u 2.00000 1.49702 0.24107 -1.13613 -1.94188 -1.77091 -0.70921 0.70921 1.77091 1.94188 1.13613 -0.24107 -1.49702 -2.00000 0.00000 0.00000 -2.00000 1.49702 0.24107 -1.13613 -1.94188 -1.77091 -0.70921 0.70921 1.77091 1.94188 1.13613 -0.24107 -1.49702 2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4u 2.00000 1.13613 -0.70921 -1.94188 -1.49702 0.24107 1.77091 1.77091 0.24107 -1.49702 -1.94188 -0.70921 1.13613 2.00000 0.00000 0.00000 -2.00000 -1.13613 0.70921 1.94188 1.49702 -0.24107 -1.77091 -1.77091 -0.24107 1.49702 1.94188 0.70921 -1.13613 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ..TT....... ............. E5u 2.00000 0.70921 -1.49702 -1.77091 0.24107 1.94188 1.13613 -1.13613 -1.94188 -0.24107 1.77091 1.49702 -0.70921 -2.00000 0.00000 0.00000 -2.00000 0.70921 -1.49702 -1.77091 0.24107 1.94188 1.13613 -1.13613 -1.94188 -0.24107 1.77091 1.49702 -0.70921 2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ............. E6u 2.00000 0.24107 -1.94188 -0.70921 1.77091 1.13613 -1.49702 -1.49702 1.13613 1.77091 -0.70921 -1.94188 0.24107 2.00000 0.00000 0.00000 -2.00000 -0.24107 1.94188 0.70921 -1.77091 -1.13613 1.49702 1.49702 -1.13613 -1.77091 0.70921 1.94188 -0.24107 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E7u 2.00000 -0.24107 -1.94188 0.70921 1.77091 -1.13613 -1.49702 1.49702 1.13613 -1.77091 -0.70921 1.94188 0.24107 -2.00000 0.00000 0.00000 -2.00000 -0.24107 -1.94188 0.70921 1.77091 -1.13613 -1.49702 1.49702 1.13613 -1.77091 -0.70921 1.94188 0.24107 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8u 2.00000 -0.70921 -1.49702 1.77091 0.24107 -1.94188 1.13613 1.13613 -1.94188 0.24107 1.77091 -1.49702 -0.70921 2.00000 0.00000 0.00000 -2.00000 0.70921 1.49702 -1.77091 -0.24107 1.94188 -1.13613 -1.13613 1.94188 -0.24107 -1.77091 1.49702 0.70921 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9u 2.00000 -1.13613 -0.70921 1.94188 -1.49702 -0.24107 1.77091 -1.77091 0.24107 1.49702 -1.94188 0.70921 1.13613 -2.00000 0.00000 0.00000 -2.00000 -1.13613 -0.70921 1.94188 -1.49702 -0.24107 1.77091 -1.77091 0.24107 1.49702 -1.94188 0.70921 1.13613 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10u 2.00000 -1.49702 0.24107 1.13613 -1.94188 1.77091 -0.70921 -0.70921 1.77091 -1.94188 1.13613 0.24107 -1.49702 2.00000 0.00000 0.00000 -2.00000 1.49702 -0.24107 -1.13613 1.94188 -1.77091 0.70921 0.70921 -1.77091 1.94188 -1.13613 -0.24107 1.49702 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11u 2.00000 -1.77091 1.13613 -0.24107 -0.70921 1.49702 -1.94188 1.94188 -1.49702 0.70921 0.24107 -1.13613 1.77091 -2.00000 0.00000 0.00000 -2.00000 -1.77091 1.13613 -0.24107 -0.70921 1.49702 -1.94188 1.94188 -1.49702 0.70921 0.24107 -1.13613 1.77091 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12u 2.00000 -1.94188 1.77091 -1.49702 1.13613 -0.70921 0.24107 0.24107 -0.70921 1.13613 -1.49702 1.77091 -1.94188 2.00000 0.00000 0.00000 -2.00000 1.94188 -1.77091 1.49702 -1.13613 0.70921 -0.24107 -0.24107 0.70921 -1.13613 1.49702 -1.77091 1.94188 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.941883634852 = 2*cos(2*π/26) = 2*cos(π/13) 1.770912051306 = 2*cos(4*π/26) = 2*cos(2*π/13) 1.497021496342 = 2*cos(6*π/26) = 2*cos(3*π/13) 1.136129493462 = 2*cos(8*π/26) = 2*cos(4*π/13) 0.709209774085 = 2*cos(10*π/26) = 2*cos(5*π/13) 0.241073360511 = 2*cos(12*π/26) = 2*cos(6*π/13) Symmetry of Rotations and Cartesian products A1g d+g+i+k+m z2, z4, z6 A2g R Rz 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 E10g 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 D26h point group is 104, and the order of the principal axis (C26) is 26. The group has 32 irreducible representations. β The D26h point group is generated by three symmetry elements that are canonically chosen C26, 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 S26 can be chosen, together with σh or C2 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′ has the x axis as a member, while the y axis is a member of the C2″ set. 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 has the xz plane as a member, while the yz plane is a member of the σd set. ε The lowest nonvanishing multipole moment in D26h 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 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. All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.
| D24h | ||
| D25h | ||
| C26 C26v C26h D26 | D26h | D26d S26 |
| D27h | ||
| D28h |
This Character Table for the D26h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.