C55v 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 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 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 ..........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 ... ..... ....... ......... ........... ............. 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 p+d+f+g+h+i+j+k+l+m z, z2, z3, z4, z5, z6 A2 R Rz 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 C55v point group is 110, and the order of the principal axis (C55) is 55. The group has 29 irreducible representations. β The C55v point group is isomorphic to D55. γ The C55v point group is generated by two symmetry elements, C55 and any σv. Also, the group may be generated from two σv planes (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The group contains one set of symmetry planes σv intersecting in the principal (z) axis. The xz plane (but not the yz plane) is a member of that set. ε The lowest nonvanishing multipole moment in C55v is 2 (dipole 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. For this group, however, some of the irrational characters cannot be expressed by a closed algebraic form using real numbers only.
| C53v | ||
| C54v | ||
| C55 | C55v | C55h D55 D55h D55d |
| C56v | ||
| C57v |
This Character Table for the C55v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.