D21d E 2 C21 2 C21^2 2 C7 2 C21^4 2 C21^5 2 C7^2 2 C3 2 C21^8 2 C7^3 2 C21^10 21 C2' i 2 S42 2 S14 2 S42^5 2 S6 2 S14^3 2 S42^11 2 S42^13 2 S14^5 2 S42^17 2 S42^19 21 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 ... ... ....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 ..T ... ..... ....... ......... ........... ............. E1g 2.00000 1.91115 1.65248 1.24698 0.73068 0.14946 -0.44504 -1.00000 -1.46610 -1.80194 -1.97766 0.00000 2.00000 -1.97766 -1.80194 -1.46610 -1.00000 -0.44504 0.14946 0.73068 1.24698 1.65248 1.91115 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g 2.00000 1.65248 0.73068 -0.44504 -1.46610 -1.97766 -1.80194 -1.00000 0.14946 1.24698 1.91115 0.00000 2.00000 1.91115 1.24698 0.14946 -1.00000 -1.80194 -1.97766 -1.46610 -0.44504 0.73068 1.65248 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3g 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 0.00000 2.00000 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g 2.00000 0.73068 -1.46610 -1.80194 0.14946 1.91115 1.24698 -1.00000 -1.97766 -0.44504 1.65248 0.00000 2.00000 1.65248 -0.44504 -1.97766 -1.00000 1.24698 1.91115 0.14946 -1.80194 -1.46610 0.73068 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5g 2.00000 0.14946 -1.97766 -0.44504 1.91115 0.73068 -1.80194 -1.00000 1.65248 1.24698 -1.46610 0.00000 2.00000 -1.46610 1.24698 1.65248 -1.00000 -1.80194 0.73068 1.91115 -0.44504 -1.97766 0.14946 0.00000 ... ... ..... ....... ......... ........... ..TT......... E6g 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 0.00000 2.00000 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 0.00000 ... ... ..... ....... ......... ........... TT........... E7g 2.00000 -1.00000 -1.00000 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 2.00000 -1.00000 -1.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8g 2.00000 -1.46610 0.14946 1.24698 -1.97766 1.65248 -0.44504 -1.00000 1.91115 -1.80194 0.73068 0.00000 2.00000 0.73068 -1.80194 1.91115 -1.00000 -0.44504 1.65248 -1.97766 1.24698 0.14946 -1.46610 0.00000 ... ... ..... ....... ......... ........... ............. E9g 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 0.00000 2.00000 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 0.00000 ... ... ..... ....... ......... ........... ............. E10g 2.00000 -1.97766 1.91115 -1.80194 1.65248 -1.46610 1.24698 -1.00000 0.73068 -0.44504 0.14946 0.00000 2.00000 0.14946 -0.44504 0.73068 -1.00000 1.24698 -1.46610 1.65248 -1.80194 1.91115 -1.97766 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 ... ... ..... ....... ......... ........... ............. 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 ... ..T ..... ......T ......... ..........T ............. E1u 2.00000 1.91115 1.65248 1.24698 0.73068 0.14946 -0.44504 -1.00000 -1.46610 -1.80194 -1.97766 0.00000 -2.00000 1.97766 1.80194 1.46610 1.00000 0.44504 -0.14946 -0.73068 -1.24698 -1.65248 -1.91115 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2u 2.00000 1.65248 0.73068 -0.44504 -1.46610 -1.97766 -1.80194 -1.00000 0.14946 1.24698 1.91115 0.00000 -2.00000 -1.91115 -1.24698 -0.14946 1.00000 1.80194 1.97766 1.46610 0.44504 -0.73068 -1.65248 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E3u 2.00000 1.24698 -0.44504 -1.80194 -1.80194 -0.44504 1.24698 2.00000 1.24698 -0.44504 -1.80194 0.00000 -2.00000 1.80194 0.44504 -1.24698 -2.00000 -1.24698 0.44504 1.80194 1.80194 0.44504 -1.24698 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4u 2.00000 0.73068 -1.46610 -1.80194 0.14946 1.91115 1.24698 -1.00000 -1.97766 -0.44504 1.65248 0.00000 -2.00000 -1.65248 0.44504 1.97766 1.00000 -1.24698 -1.91115 -0.14946 1.80194 1.46610 -0.73068 0.00000 ... ... ..... ....... ......... ..TT....... ............. E5u 2.00000 0.14946 -1.97766 -0.44504 1.91115 0.73068 -1.80194 -1.00000 1.65248 1.24698 -1.46610 0.00000 -2.00000 1.46610 -1.24698 -1.65248 1.00000 1.80194 -0.73068 -1.91115 0.44504 1.97766 -0.14946 0.00000 ... ... ..... ....... ......... TT......... ............. E6u 2.00000 -0.44504 -1.80194 1.24698 1.24698 -1.80194 -0.44504 2.00000 -0.44504 -1.80194 1.24698 0.00000 -2.00000 -1.24698 1.80194 0.44504 -2.00000 0.44504 1.80194 -1.24698 -1.24698 1.80194 0.44504 0.00000 ... ... ..... ....... ......... ........... ............. E7u 2.00000 -1.00000 -1.00000 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 -2.00000 1.00000 1.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8u 2.00000 -1.46610 0.14946 1.24698 -1.97766 1.65248 -0.44504 -1.00000 1.91115 -1.80194 0.73068 0.00000 -2.00000 -0.73068 1.80194 -1.91115 1.00000 0.44504 -1.65248 1.97766 -1.24698 -0.14946 1.46610 0.00000 ... ... ..... ....... ......... ........... ............. E9u 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 0.00000 -2.00000 0.44504 -1.24698 1.80194 -2.00000 1.80194 -1.24698 0.44504 0.44504 -1.24698 1.80194 0.00000 ... ... ..... ....... ......... ........... ............. E10u 2.00000 -1.97766 1.91115 -1.80194 1.65248 -1.46610 1.24698 -1.00000 0.73068 -0.44504 0.14946 0.00000 -2.00000 -0.14946 0.44504 -0.73068 1.00000 -1.24698 1.46610 -1.65248 1.80194 -1.91115 1.97766 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.977661652450 = 2*cos(2*π/42) = 2*cos(π/21) 1.911145611572 = 2*cos(4*π/42) = 2*cos(2*π/21) 1.801937735805 = 2*cos(6*π/42) = 2*cos(π/7) 1.652477548632 = 2*cos(8*π/42) = 2*cos(4*π/21) 1.466103743660 = 2*cos(10*π/42) = 2*cos(5*π/21) 1.246979603717 = 2*cos(12*π/42) = 2*cos(2*π/7) 0.730682048733 = 2*cos(16*π/42) = 2*cos(8*π/21) 0.445041867913 = 2*cos(18*π/42) = 2*cos(3*π/7) 0.149460187173 = 2*cos(20*π/42) = 2*cos(10*π/21) 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 D21d point group is 84, and the order of the principal axis (S42) is 42. The group has 24 irreducible representations. β The D21d point group is isomorphic to D21h, C42v and D42. γ The D21d point group is generated by two symmetry elements, S42 and either a perpendicular C2′ or a vertical σd. Also, the group may be generated from a C2′ plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe). The canonical choice, however, is to use redundant generators: C21, C2′ and i. 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. Reversely, the single set of symmetry planes denoted σd contains the yz plane but not the xz plane. ε The lowest nonvanishing multipole moment in D21d 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.
| D19d | ||
| D20d | ||
| C21 C21v C21h D21 D21h | D21d | |
| D22d | ||
| D23d |
This Character Table for the D21d point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.