S58 E 2 C29 2 C29^2 2 C29^3 2 C29^4 2 C29^5 2 C29^6 2 C29^7 2 C29^8 2 C29^9 2 C29^10 2 C29^11 2 C29^12 2 C29^13 2 C29^14 i 2 S58 2 S58^3 2 S58^5 2 S58^7 2 S58^9 2 S58^11 2 S58^13 2 S58^15 2 S58^17 2 S58^19 2 S58^21 2 S58^23 2 S58^25 2 S58^27 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> Ag 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 E1g * 2.00000 1.95324 1.81515 1.59219 1.29477 0.93682 0.53506 0.10828 -0.32356 -0.74028 -1.12237 -1.45199 -1.71371 -1.89531 -1.98828 2.00000 -1.98828 -1.89531 -1.71371 -1.45199 -1.12237 -0.74028 -0.32356 0.10828 0.53506 0.93682 1.29477 1.59219 1.81515 1.95324 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g * 2.00000 1.81515 1.29477 0.53506 -0.32356 -1.12237 -1.71371 -1.98828 -1.89531 -1.45199 -0.74028 0.10828 0.93682 1.59219 1.95324 2.00000 1.95324 1.59219 0.93682 0.10828 -0.74028 -1.45199 -1.89531 -1.98828 -1.71371 -1.12237 -0.32356 0.53506 1.29477 1.81515 ... ... TT... ....... ....TT... ........... ........TT... E3g * 2.00000 1.59219 0.53506 -0.74028 -1.71371 -1.98828 -1.45199 -0.32356 0.93682 1.81515 1.95324 1.29477 0.10828 -1.12237 -1.89531 2.00000 -1.89531 -1.12237 0.10828 1.29477 1.95324 1.81515 0.93682 -0.32356 -1.45199 -1.98828 -1.71371 -0.74028 0.53506 1.59219 ... ... ..... ....... ..TT..... ........... ......TT..... E4g * 2.00000 1.29477 -0.32356 -1.71371 -1.89531 -0.74028 0.93682 1.95324 1.59219 0.10828 -1.45199 -1.98828 -1.12237 0.53506 1.81515 2.00000 1.81515 0.53506 -1.12237 -1.98828 -1.45199 0.10828 1.59219 1.95324 0.93682 -0.74028 -1.89531 -1.71371 -0.32356 1.29477 ... ... ..... ....... TT....... ........... ....TT....... E5g * 2.00000 0.93682 -1.12237 -1.98828 -0.74028 1.29477 1.95324 0.53506 -1.45199 -1.89531 -0.32356 1.59219 1.81515 0.10828 -1.71371 2.00000 -1.71371 0.10828 1.81515 1.59219 -0.32356 -1.89531 -1.45199 0.53506 1.95324 1.29477 -0.74028 -1.98828 -1.12237 0.93682 ... ... ..... ....... ......... ........... ..TT......... E6g * 2.00000 0.53506 -1.71371 -1.45199 0.93682 1.95324 0.10828 -1.89531 -1.12237 1.29477 1.81515 -0.32356 -1.98828 -0.74028 1.59219 2.00000 1.59219 -0.74028 -1.98828 -0.32356 1.81515 1.29477 -1.12237 -1.89531 0.10828 1.95324 0.93682 -1.45199 -1.71371 0.53506 ... ... ..... ....... ......... ........... TT........... E7g * 2.00000 0.10828 -1.98828 -0.32356 1.95324 0.53506 -1.89531 -0.74028 1.81515 0.93682 -1.71371 -1.12237 1.59219 1.29477 -1.45199 2.00000 -1.45199 1.29477 1.59219 -1.12237 -1.71371 0.93682 1.81515 -0.74028 -1.89531 0.53506 1.95324 -0.32356 -1.98828 0.10828 ... ... ..... ....... ......... ........... ............. E8g * 2.00000 -0.32356 -1.89531 0.93682 1.59219 -1.45199 -1.12237 1.81515 0.53506 -1.98828 0.10828 1.95324 -0.74028 -1.71371 1.29477 2.00000 1.29477 -1.71371 -0.74028 1.95324 0.10828 -1.98828 0.53506 1.81515 -1.12237 -1.45199 1.59219 0.93682 -1.89531 -0.32356 ... ... ..... ....... ......... ........... ............. E9g * 2.00000 -0.74028 -1.45199 1.81515 0.10828 -1.89531 1.29477 0.93682 -1.98828 0.53506 1.59219 -1.71371 -0.32356 1.95324 -1.12237 2.00000 -1.12237 1.95324 -0.32356 -1.71371 1.59219 0.53506 -1.98828 0.93682 1.29477 -1.89531 0.10828 1.81515 -1.45199 -0.74028 ... ... ..... ....... ......... ........... ............. E10g * 2.00000 -1.12237 -0.74028 1.95324 -1.45199 -0.32356 1.81515 -1.71371 0.10828 1.59219 -1.89531 0.53506 1.29477 -1.98828 0.93682 2.00000 0.93682 -1.98828 1.29477 0.53506 -1.89531 1.59219 0.10828 -1.71371 1.81515 -0.32356 -1.45199 1.95324 -0.74028 -1.12237 ... ... ..... ....... ......... ........... ............. E11g * 2.00000 -1.45199 0.10828 1.29477 -1.98828 1.59219 -0.32356 -1.12237 1.95324 -1.71371 0.53506 0.93682 -1.89531 1.81515 -0.74028 2.00000 -0.74028 1.81515 -1.89531 0.93682 0.53506 -1.71371 1.95324 -1.12237 -0.32356 1.59219 -1.98828 1.29477 0.10828 -1.45199 ... ... ..... ....... ......... ........... ............. E12g * 2.00000 -1.71371 0.93682 0.10828 -1.12237 1.81515 -1.98828 1.59219 -0.74028 -0.32356 1.29477 -1.89531 1.95324 -1.45199 0.53506 2.00000 0.53506 -1.45199 1.95324 -1.89531 1.29477 -0.32356 -0.74028 1.59219 -1.98828 1.81515 -1.12237 0.10828 0.93682 -1.71371 ... ... ..... ....... ......... ........... ............. E13g * 2.00000 -1.89531 1.59219 -1.12237 0.53506 0.10828 -0.74028 1.29477 -1.71371 1.95324 -1.98828 1.81515 -1.45199 0.93682 -0.32356 2.00000 -0.32356 0.93682 -1.45199 1.81515 -1.98828 1.95324 -1.71371 1.29477 -0.74028 0.10828 0.53506 -1.12237 1.59219 -1.89531 ... ... ..... ....... ......... ........... ............. E14g * 2.00000 -1.98828 1.95324 -1.89531 1.81515 -1.71371 1.59219 -1.45199 1.29477 -1.12237 0.93682 -0.74028 0.53506 -0.32356 0.10828 2.00000 0.10828 -0.32356 0.53506 -0.74028 0.93682 -1.12237 1.29477 -1.45199 1.59219 -1.71371 1.81515 -1.89531 1.95324 -1.98828 ... ... ..... ....... ......... ........... ............. Au 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -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 1.95324 1.81515 1.59219 1.29477 0.93682 0.53506 0.10828 -0.32356 -0.74028 -1.12237 -1.45199 -1.71371 -1.89531 -1.98828 -2.00000 1.98828 1.89531 1.71371 1.45199 1.12237 0.74028 0.32356 -0.10828 -0.53506 -0.93682 -1.29477 -1.59219 -1.81515 -1.95324 ... TT. ..... ....TT. ......... ........TT. ............. E2u * 2.00000 1.81515 1.29477 0.53506 -0.32356 -1.12237 -1.71371 -1.98828 -1.89531 -1.45199 -0.74028 0.10828 0.93682 1.59219 1.95324 -2.00000 -1.95324 -1.59219 -0.93682 -0.10828 0.74028 1.45199 1.89531 1.98828 1.71371 1.12237 0.32356 -0.53506 -1.29477 -1.81515 ... ... ..... ..TT... ......... ......TT... ............. E3u * 2.00000 1.59219 0.53506 -0.74028 -1.71371 -1.98828 -1.45199 -0.32356 0.93682 1.81515 1.95324 1.29477 0.10828 -1.12237 -1.89531 -2.00000 1.89531 1.12237 -0.10828 -1.29477 -1.95324 -1.81515 -0.93682 0.32356 1.45199 1.98828 1.71371 0.74028 -0.53506 -1.59219 ... ... ..... TT..... ......... ....TT..... ............. E4u * 2.00000 1.29477 -0.32356 -1.71371 -1.89531 -0.74028 0.93682 1.95324 1.59219 0.10828 -1.45199 -1.98828 -1.12237 0.53506 1.81515 -2.00000 -1.81515 -0.53506 1.12237 1.98828 1.45199 -0.10828 -1.59219 -1.95324 -0.93682 0.74028 1.89531 1.71371 0.32356 -1.29477 ... ... ..... ....... ......... ..TT....... ............. E5u * 2.00000 0.93682 -1.12237 -1.98828 -0.74028 1.29477 1.95324 0.53506 -1.45199 -1.89531 -0.32356 1.59219 1.81515 0.10828 -1.71371 -2.00000 1.71371 -0.10828 -1.81515 -1.59219 0.32356 1.89531 1.45199 -0.53506 -1.95324 -1.29477 0.74028 1.98828 1.12237 -0.93682 ... ... ..... ....... ......... TT......... ............. E6u * 2.00000 0.53506 -1.71371 -1.45199 0.93682 1.95324 0.10828 -1.89531 -1.12237 1.29477 1.81515 -0.32356 -1.98828 -0.74028 1.59219 -2.00000 -1.59219 0.74028 1.98828 0.32356 -1.81515 -1.29477 1.12237 1.89531 -0.10828 -1.95324 -0.93682 1.45199 1.71371 -0.53506 ... ... ..... ....... ......... ........... ............. E7u * 2.00000 0.10828 -1.98828 -0.32356 1.95324 0.53506 -1.89531 -0.74028 1.81515 0.93682 -1.71371 -1.12237 1.59219 1.29477 -1.45199 -2.00000 1.45199 -1.29477 -1.59219 1.12237 1.71371 -0.93682 -1.81515 0.74028 1.89531 -0.53506 -1.95324 0.32356 1.98828 -0.10828 ... ... ..... ....... ......... ........... ............. E8u * 2.00000 -0.32356 -1.89531 0.93682 1.59219 -1.45199 -1.12237 1.81515 0.53506 -1.98828 0.10828 1.95324 -0.74028 -1.71371 1.29477 -2.00000 -1.29477 1.71371 0.74028 -1.95324 -0.10828 1.98828 -0.53506 -1.81515 1.12237 1.45199 -1.59219 -0.93682 1.89531 0.32356 ... ... ..... ....... ......... ........... ............. E9u * 2.00000 -0.74028 -1.45199 1.81515 0.10828 -1.89531 1.29477 0.93682 -1.98828 0.53506 1.59219 -1.71371 -0.32356 1.95324 -1.12237 -2.00000 1.12237 -1.95324 0.32356 1.71371 -1.59219 -0.53506 1.98828 -0.93682 -1.29477 1.89531 -0.10828 -1.81515 1.45199 0.74028 ... ... ..... ....... ......... ........... ............. E10u * 2.00000 -1.12237 -0.74028 1.95324 -1.45199 -0.32356 1.81515 -1.71371 0.10828 1.59219 -1.89531 0.53506 1.29477 -1.98828 0.93682 -2.00000 -0.93682 1.98828 -1.29477 -0.53506 1.89531 -1.59219 -0.10828 1.71371 -1.81515 0.32356 1.45199 -1.95324 0.74028 1.12237 ... ... ..... ....... ......... ........... ............. E11u * 2.00000 -1.45199 0.10828 1.29477 -1.98828 1.59219 -0.32356 -1.12237 1.95324 -1.71371 0.53506 0.93682 -1.89531 1.81515 -0.74028 -2.00000 0.74028 -1.81515 1.89531 -0.93682 -0.53506 1.71371 -1.95324 1.12237 0.32356 -1.59219 1.98828 -1.29477 -0.10828 1.45199 ... ... ..... ....... ......... ........... ............. E12u * 2.00000 -1.71371 0.93682 0.10828 -1.12237 1.81515 -1.98828 1.59219 -0.74028 -0.32356 1.29477 -1.89531 1.95324 -1.45199 0.53506 -2.00000 -0.53506 1.45199 -1.95324 1.89531 -1.29477 0.32356 0.74028 -1.59219 1.98828 -1.81515 1.12237 -0.10828 -0.93682 1.71371 ... ... ..... ....... ......... ........... ............. E13u * 2.00000 -1.89531 1.59219 -1.12237 0.53506 0.10828 -0.74028 1.29477 -1.71371 1.95324 -1.98828 1.81515 -1.45199 0.93682 -0.32356 -2.00000 0.32356 -0.93682 1.45199 -1.81515 1.98828 -1.95324 1.71371 -1.29477 0.74028 -0.10828 -0.53506 1.12237 -1.59219 1.89531 ... ... ..... ....... ......... ........... ............. E14u * 2.00000 -1.98828 1.95324 -1.89531 1.81515 -1.71371 1.59219 -1.45199 1.29477 -1.12237 0.93682 -0.74028 0.53506 -0.32356 0.10828 -2.00000 -0.10828 0.32356 -0.53506 0.74028 -0.93682 1.12237 -1.29477 1.45199 -1.59219 1.71371 -1.81515 1.89531 -1.95324 1.98828 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.988275914309 = 2*cos(2*π/58) = 2*cos(π/29) 1.953241111420 = 2*cos(4*π/58) = 2*cos(2*π/29) 1.895306342366 = 2*cos(6*π/58) = 2*cos(3*π/29) 1.815150839342 = 2*cos(8*π/58) = 2*cos(4*π/29) 1.713714352335 = 2*cos(10*π/58) = 2*cos(5*π/29) 1.592186131411 = 2*cos(12*π/58) = 2*cos(6*π/29) 1.451990983846 = 2*cos(14*π/58) = 2*cos(7*π/29) 1.294772569564 = 2*cos(16*π/58) = 2*cos(8*π/29) 1.122374130725 = 2*cos(18*π/58) = 2*cos(9*π/29) 0.936816881400 = 2*cos(20*π/58) = 2*cos(10*π/29) 0.740276310680 = 2*cos(22*π/58) = 2*cos(11*π/29) 0.535056677058 = 2*cos(24*π/58) = 2*cos(12*π/29) 0.323563993106 = 2*cos(26*π/58) = 2*cos(13*π/29) 0.108277817171 = 2*cos(28*π/58) = 2*cos(14*π/29) Symmetry of Rotations and Cartesian products Ag R+d+g+i+k+m Rz, z2, z4, z6 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 Au 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 S58 point group is 58, and the order of the principal axis (S58) is 58. The group has 30 irreducible representations. β The S58 point group could also be named C29i, because the S58 axis is identical to a roto-inversion axis of order 29. γ The S58 point group is isomorphic to C58 and C29h. δ The S58 point group is generated by one single symmetry element, S58. Therefore, it is a cyclic group. ε The lowest nonvanishing multipole moment in S58 is 4 (quadrupole moment). ζ This is an Abelian point group (the commutative law holds between all symmetry operations). The S58 group is Abelian because it contains only one symmetry element, all the powers of which necessarily commute (sufficient condition). In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. η Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 56 cases have been combined into 28 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 28 pairs of left and right rotations have been combined into one two-membered pseudo-class each. θ The 28 reducible “E” representations almost behave like true irreducible representations. Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table. This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula. ι 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, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.
| S54 | ||
| S56 | ||
| C58 C58v C58h D58 D58h D58d | S58 | |
| S60 | ||
| S62 |
This Character Table for the S58 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.