D55 E 2 C55 2 C55^2 2 C55^3 2 C55^4 2 C11 2 C55^6 2 C55^7 2 C55^8 2 C55^9 2 C11^2 2 C5 2 C55^12 2 C55^13 2 C55^14 2 C11^3 2 C55^16 2 C55^17 2 C55^18 2 C55^19 2 C11^4 2 C55^21 2 C5^2 2 C55^23 2 C55^24 2 C11^5 2 C55^26 2 C55^27 55 C2' <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 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 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 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 ......... ..........T ............. E1 2.00000 1.98696 1.94802 1.88369 1.79480 1.68251 1.54828 1.39388 1.22130 1.03279 0.83083 0.61803 0.39718 0.17115 -0.05711 -0.28463 -0.50844 -0.72562 -0.93333 -1.12889 -1.30972 -1.47348 -1.61803 -1.74149 -1.84225 -1.91899 -1.97071 -1.99674 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.94802 1.79480 1.54828 1.22130 0.83083 0.39718 -0.05711 -0.50844 -0.93333 -1.30972 -1.61803 -1.84225 -1.97071 -1.99674 -1.91899 -1.74149 -1.47348 -1.12889 -0.72562 -0.28463 0.17115 0.61803 1.03279 1.39388 1.68251 1.88369 1.98696 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.88369 1.54828 1.03279 0.39718 -0.28463 -0.93333 -1.47348 -1.84225 -1.99674 -1.91899 -1.61803 -1.12889 -0.50844 0.17115 0.83083 1.39388 1.79480 1.98696 1.94802 1.68251 1.22130 0.61803 -0.05711 -0.72562 -1.30972 -1.74149 -1.97071 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.79480 1.22130 0.39718 -0.50844 -1.30972 -1.84225 -1.99674 -1.74149 -1.12889 -0.28463 0.61803 1.39388 1.88369 1.98696 1.68251 1.03279 0.17115 -0.72562 -1.47348 -1.91899 -1.97071 -1.61803 -0.93333 -0.05711 0.83083 1.54828 1.94802 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.54828 0.39718 -0.93333 -1.84225 -1.91899 -1.12889 0.17115 1.39388 1.98696 1.68251 0.61803 -0.72562 -1.74149 -1.97071 -1.30972 -0.05711 1.22130 1.94802 1.79480 0.83083 -0.50844 -1.61803 -1.99674 -1.47348 -0.28463 1.03279 1.88369 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.39388 -0.05711 -1.47348 -1.99674 -1.30972 0.17115 1.54828 1.98696 1.22130 -0.28463 -1.61803 -1.97071 -1.12889 0.39718 1.68251 1.94802 1.03279 -0.50844 -1.74149 -1.91899 -0.93333 0.61803 1.79480 1.88369 0.83083 -0.72562 -1.84225 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.22130 -0.50844 -1.84225 -1.74149 -0.28463 1.39388 1.98696 1.03279 -0.72562 -1.91899 -1.61803 -0.05711 1.54828 1.94802 0.83083 -0.93333 -1.97071 -1.47348 0.17115 1.68251 1.88369 0.61803 -1.12889 -1.99674 -1.30972 0.39718 1.79480 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.03279 -0.93333 -1.99674 -1.12889 0.83083 1.98696 1.22130 -0.72562 -1.97071 -1.30972 0.61803 1.94802 1.39388 -0.50844 -1.91899 -1.47348 0.39718 1.88369 1.54828 -0.28463 -1.84225 -1.61803 0.17115 1.79480 1.68251 -0.05711 -1.74149 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 -1.61803 0.61803 2.00000 0.61803 -1.61803 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 0.39718 -1.84225 -1.12889 1.39388 1.68251 -0.72562 -1.97071 -0.05711 1.94802 0.83083 -1.61803 -1.47348 1.03279 1.88369 -0.28463 -1.99674 -0.50844 1.79480 1.22130 -1.30972 -1.74149 0.61803 1.98696 0.17115 -1.91899 -0.93333 1.54828 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 0.17115 -1.97071 -0.50844 1.88369 0.83083 -1.74149 -1.12889 1.54828 1.39388 -1.30972 -1.61803 1.03279 1.79480 -0.72562 -1.91899 0.39718 1.98696 -0.05711 -1.99674 -0.28463 1.94802 0.61803 -1.84225 -0.93333 1.68251 1.22130 -1.47348 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 -0.05711 -1.99674 0.17115 1.98696 -0.28463 -1.97071 0.39718 1.94802 -0.50844 -1.91899 0.61803 1.88369 -0.72562 -1.84225 0.83083 1.79480 -0.93333 -1.74149 1.03279 1.68251 -1.12889 -1.61803 1.22130 1.54828 -1.30972 -1.47348 1.39388 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 -0.50844 -1.74149 1.39388 1.03279 -1.91899 -0.05711 1.94802 -0.93333 -1.47348 1.68251 0.61803 -1.99674 0.39718 1.79480 -1.30972 -1.12889 1.88369 0.17115 -1.97071 0.83083 1.54828 -1.61803 -0.72562 1.98696 -0.28463 -1.84225 1.22130 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 -0.72562 -1.47348 1.79480 0.17115 -1.91899 1.22130 1.03279 -1.97071 0.39718 1.68251 -1.61803 -0.50844 1.98696 -0.93333 -1.30972 1.88369 -0.05711 -1.84225 1.39388 0.83083 -1.99674 0.61803 1.54828 -1.74149 -0.28463 1.94802 -1.12889 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 -0.93333 -1.12889 1.98696 -0.72562 -1.30972 1.94802 -0.50844 -1.47348 1.88369 -0.28463 -1.61803 1.79480 -0.05711 -1.74149 1.68251 0.17115 -1.84225 1.54828 0.39718 -1.91899 1.39388 0.61803 -1.97071 1.22130 0.83083 -1.99674 1.03279 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 -1.12889 -0.72562 1.94802 -1.47348 -0.28463 1.79480 -1.74149 0.17115 1.54828 -1.91899 0.61803 1.22130 -1.99674 1.03279 0.83083 -1.97071 1.39388 0.39718 -1.84225 1.68251 -0.05711 -1.61803 1.88369 -0.50844 -1.30972 1.98696 -0.93333 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 -1.47348 0.17115 1.22130 -1.97071 1.68251 -0.50844 -0.93333 1.88369 -1.84225 0.83083 0.61803 -1.74149 1.94802 -1.12889 -0.28463 1.54828 -1.99674 1.39388 -0.05711 -1.30972 1.98696 -1.61803 0.39718 1.03279 -1.91899 1.79480 -0.72562 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.61803 -1.61803 2.00000 -1.61803 0.61803 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -1.74149 1.03279 -0.05711 -0.93333 1.68251 -1.99674 1.79480 -1.12889 0.17115 0.83083 -1.61803 1.98696 -1.84225 1.22130 -0.28463 -0.72562 1.54828 -1.97071 1.88369 -1.30972 0.39718 0.61803 -1.47348 1.94802 -1.91899 1.39388 -0.50844 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 -1.84225 1.39388 -0.72562 -0.05711 0.83083 -1.47348 1.88369 -1.99674 1.79480 -1.30972 0.61803 0.17115 -0.93333 1.54828 -1.91899 1.98696 -1.74149 1.22130 -0.50844 -0.28463 1.03279 -1.61803 1.94802 -1.97071 1.68251 -1.12889 0.39718 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -1.97071 1.88369 -1.74149 1.54828 -1.30972 1.03279 -0.72562 0.39718 -0.05711 -0.28463 0.61803 -0.93333 1.22130 -1.47348 1.68251 -1.84225 1.94802 -1.99674 1.98696 -1.91899 1.79480 -1.61803 1.39388 -1.12889 0.83083 -0.50844 0.17115 0.00000 ... ... ..... ....... ......... ........... ............. E27 2.00000 -1.99674 1.98696 -1.97071 1.94802 -1.91899 1.88369 -1.84225 1.79480 -1.74149 1.68251 -1.61803 1.54828 -1.47348 1.39388 -1.30972 1.22130 -1.12889 1.03279 -0.93333 0.83083 -0.72562 0.61803 -0.50844 0.39718 -0.28463 0.17115 -0.05711 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.986963470697 = 2*cos(2*π/55) 1.948023833885 = 2*cos(4*π/55) 1.883688727279 = 2*cos(6*π/55) 1.794796857383 = 2*cos(8*π/55) 1.682507065662 = 2*cos(10*π/55) = 2*cos(2*π/11) 1.548283221278 = 2*cos(12*π/55) 1.393875137311 = 2*cos(14*π/55) 1.221295759271 = 2*cos(16*π/55) 1.032794923278 = 2*cos(18*π/55) 0.830830026004 = 2*cos(20*π/55) = 2*cos(4*π/11) 0.618033988750 = 2*cos(22*π/55) = 2*cos(2*π/5) = (√5−1)/2 0.397180933291 = 2*cos(24*π/55) 0.171150016958 = 2*cos(26*π/55) -0.057112101587 = 2*cos(28*π/55) -0.284629676547 = 2*cos(30*π/55) = 2*cos(6*π/11) -0.508436668387 = 2*cos(32*π/55) -0.725615410701 = 2*cos(34*π/55) -0.933334646451 = 2*cos(36*π/55) -1.128886437734 = 2*cos(38*π/55) -1.309721467891 = 2*cos(40*π/55) = 2*cos(8*π/11) -1.473482275753 = 2*cos(42*π/55) -1.618033988750 = 2*cos(44*π/55) = 2*cos(4*π/5) = −(√5+1)/2 -1.741492154240 = 2*cos(46*π/55) -1.842247306230 = 2*cos(48*π/55) -1.918985947229 = 2*cos(50*π/55) = 2*cos(10*π/11) -1.970707671695 = 2*cos(52*π/55) -1.996738207852 = 2*cos(54*π/55) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z2, z4, z6 A2 R+p+f+h+j+l Rz, z, z3, z5 E1 R+p+d+f+g+h+i+j+k+l+m {Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {xz3, yz3}, {xz4, yz4}, {xz5, yz5} E2 d+f+g+h+i+j+k+l+m {x2−y2, xy}, {z(x2−y2), xyz}, {z2(x2−y2), xyz2}, {z3(x2−y2), xyz3}, {z4(x2−y2), xyz4} E3 f+g+h+i+j+k+l+m {x(x2−3y2), y(3x2−y2)}, {xz(x2−3y2), yz(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4 g+h+i+j+k+l+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 h+i+j+k+l+m {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6 i+j+k+l+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the D55 point group is 110, and the order of the principal axis (C55) is 55. The group has 29 irreducible representations. β The D55 point group is isomorphic to C55v. γ The D55 point group is generated by two symmetry elements, C55 and a perpendicular C2′. Also, the group may be generated from two C2′ axes (some pairs will yield smaller groups, though; choosing a minimum angle is safe). 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. ε The lowest nonvanishing multipole moment in D55 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. For this group, however, some of the irrational characters cannot be expressed by a closed algebraic form using real numbers only.
| D53 | ||
| D54 | ||
| C55 C55v C55h | D55 | D55h D55d |
| D56 | ||
| D57 |
This Character Table for the D55 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.