Character table for the S54 point group

S54      E        2 C27    2 C27^2  2 C9     2 C27^4  2 C27^5  2 C9^2   2 C27^7  2 C27^8  2 C3     2 C27^10 2 C27^11 2 C9^4   2 C27^13 i        2 S54    2 S18    2 S54^5  2 S54^7  2 S6     2 S54^11 2 S54^13 2 S54^15 2 S54^17 2 S54^19 2 S54^21 2 S54^23 2 S54^25   <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     ..T ... ....T ....... ........T ........... ............T
E1g  *  2.00000  1.94609  1.78727  1.53209  1.19432  0.79216  0.34730 -0.11629 -0.57361 -1.00000 -1.37248 -1.67098 -1.87939 -1.98648  2.00000 -1.98648 -1.87939 -1.67098 -1.37248 -1.00000 -0.57361 -0.11629  0.34730  0.79216  1.19432  1.53209  1.78727  1.94609     TT. ... ..TT. ....... ......TT. ........... ..........TT.
E2g  *  2.00000  1.78727  1.19432  0.34730 -0.57361 -1.37248 -1.87939 -1.98648 -1.67098 -1.00000 -0.11629  0.79216  1.53209  1.94609  2.00000  1.94609  1.53209  0.79216 -0.11629 -1.00000 -1.67098 -1.98648 -1.87939 -1.37248 -0.57361  0.34730  1.19432  1.78727     ... ... TT... ....... ....TT... ........... ........TT...
E3g  *  2.00000  1.53209  0.34730 -1.00000 -1.87939 -1.87939 -1.00000  0.34730  1.53209  2.00000  1.53209  0.34730 -1.00000 -1.87939  2.00000 -1.87939 -1.00000  0.34730  1.53209  2.00000  1.53209  0.34730 -1.00000 -1.87939 -1.87939 -1.00000  0.34730  1.53209     ... ... ..... ....... ..TT..... ........... ......TT.....
E4g  *  2.00000  1.19432 -0.57361 -1.87939 -1.67098 -0.11629  1.53209  1.94609  0.79216 -1.00000 -1.98648 -1.37248  0.34730  1.78727  2.00000  1.78727  0.34730 -1.37248 -1.98648 -1.00000  0.79216  1.94609  1.53209 -0.11629 -1.67098 -1.87939 -0.57361  1.19432     ... ... ..... ....... TT....... ........... ....TT.......
E5g  *  2.00000  0.79216 -1.37248 -1.87939 -0.11629  1.78727  1.53209 -0.57361 -1.98648 -1.00000  1.19432  1.94609  0.34730 -1.67098  2.00000 -1.67098  0.34730  1.94609  1.19432 -1.00000 -1.98648 -0.57361  1.53209  1.78727 -0.11629 -1.87939 -1.37248  0.79216     ... ... ..... ....... ......... ........... ..TT.........
E6g  *  2.00000  0.34730 -1.87939 -1.00000  1.53209  1.53209 -1.00000 -1.87939  0.34730  2.00000  0.34730 -1.87939 -1.00000  1.53209  2.00000  1.53209 -1.00000 -1.87939  0.34730  2.00000  0.34730 -1.87939 -1.00000  1.53209  1.53209 -1.00000 -1.87939  0.34730     ... ... ..... ....... ......... ........... TT...........
E7g  *  2.00000 -0.11629 -1.98648  0.34730  1.94609 -0.57361 -1.87939  0.79216  1.78727 -1.00000 -1.67098  1.19432  1.53209 -1.37248  2.00000 -1.37248  1.53209  1.19432 -1.67098 -1.00000  1.78727  0.79216 -1.87939 -0.57361  1.94609  0.34730 -1.98648 -0.11629     ... ... ..... ....... ......... ........... .............
E8g  *  2.00000 -0.57361 -1.67098  1.53209  0.79216 -1.98648  0.34730  1.78727 -1.37248 -1.00000  1.94609 -0.11629 -1.87939  1.19432  2.00000  1.19432 -1.87939 -0.11629  1.94609 -1.00000 -1.37248  1.78727  0.34730 -1.98648  0.79216  1.53209 -1.67098 -0.57361     ... ... ..... ....... ......... ........... .............
E9g  *  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000  2.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000     ... ... ..... ....... ......... ........... .............
E10g *  2.00000 -1.37248 -0.11629  1.53209 -1.98648  1.19432  0.34730 -1.67098  1.94609 -1.00000 -0.57361  1.78727 -1.87939  0.79216  2.00000  0.79216 -1.87939  1.78727 -0.57361 -1.00000  1.94609 -1.67098  0.34730  1.19432 -1.98648  1.53209 -0.11629 -1.37248     ... ... ..... ....... ......... ........... .............
E11g *  2.00000 -1.67098  0.79216  0.34730 -1.37248  1.94609 -1.87939  1.19432 -0.11629 -1.00000  1.78727 -1.98648  1.53209 -0.57361  2.00000 -0.57361  1.53209 -1.98648  1.78727 -1.00000 -0.11629  1.19432 -1.87939  1.94609 -1.37248  0.34730  0.79216 -1.67098     ... ... ..... ....... ......... ........... .............
E12g *  2.00000 -1.87939  1.53209 -1.00000  0.34730  0.34730 -1.00000  1.53209 -1.87939  2.00000 -1.87939  1.53209 -1.00000  0.34730  2.00000  0.34730 -1.00000  1.53209 -1.87939  2.00000 -1.87939  1.53209 -1.00000  0.34730  0.34730 -1.00000  1.53209 -1.87939     ... ... ..... ....... ......... ........... .............
E13g *  2.00000 -1.98648  1.94609 -1.87939  1.78727 -1.67098  1.53209 -1.37248  1.19432 -1.00000  0.79216 -0.57361  0.34730 -0.11629  2.00000 -0.11629  0.34730 -0.57361  0.79216 -1.00000  1.19432 -1.37248  1.53209 -1.67098  1.78727 -1.87939  1.94609 -1.98648     ... ... ..... ....... ......... ........... .............
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     ... ..T ..... ......T ......... ..........T .............
E1u  *  2.00000  1.94609  1.78727  1.53209  1.19432  0.79216  0.34730 -0.11629 -0.57361 -1.00000 -1.37248 -1.67098 -1.87939 -1.98648 -2.00000  1.98648  1.87939  1.67098  1.37248  1.00000  0.57361  0.11629 -0.34730 -0.79216 -1.19432 -1.53209 -1.78727 -1.94609     ... TT. ..... ....TT. ......... ........TT. .............
E2u  *  2.00000  1.78727  1.19432  0.34730 -0.57361 -1.37248 -1.87939 -1.98648 -1.67098 -1.00000 -0.11629  0.79216  1.53209  1.94609 -2.00000 -1.94609 -1.53209 -0.79216  0.11629  1.00000  1.67098  1.98648  1.87939  1.37248  0.57361 -0.34730 -1.19432 -1.78727     ... ... ..... ..TT... ......... ......TT... .............
E3u  *  2.00000  1.53209  0.34730 -1.00000 -1.87939 -1.87939 -1.00000  0.34730  1.53209  2.00000  1.53209  0.34730 -1.00000 -1.87939 -2.00000  1.87939  1.00000 -0.34730 -1.53209 -2.00000 -1.53209 -0.34730  1.00000  1.87939  1.87939  1.00000 -0.34730 -1.53209     ... ... ..... TT..... ......... ....TT..... .............
E4u  *  2.00000  1.19432 -0.57361 -1.87939 -1.67098 -0.11629  1.53209  1.94609  0.79216 -1.00000 -1.98648 -1.37248  0.34730  1.78727 -2.00000 -1.78727 -0.34730  1.37248  1.98648  1.00000 -0.79216 -1.94609 -1.53209  0.11629  1.67098  1.87939  0.57361 -1.19432     ... ... ..... ....... ......... ..TT....... .............
E5u  *  2.00000  0.79216 -1.37248 -1.87939 -0.11629  1.78727  1.53209 -0.57361 -1.98648 -1.00000  1.19432  1.94609  0.34730 -1.67098 -2.00000  1.67098 -0.34730 -1.94609 -1.19432  1.00000  1.98648  0.57361 -1.53209 -1.78727  0.11629  1.87939  1.37248 -0.79216     ... ... ..... ....... ......... TT......... .............
E6u  *  2.00000  0.34730 -1.87939 -1.00000  1.53209  1.53209 -1.00000 -1.87939  0.34730  2.00000  0.34730 -1.87939 -1.00000  1.53209 -2.00000 -1.53209  1.00000  1.87939 -0.34730 -2.00000 -0.34730  1.87939  1.00000 -1.53209 -1.53209  1.00000  1.87939 -0.34730     ... ... ..... ....... ......... ........... .............
E7u  *  2.00000 -0.11629 -1.98648  0.34730  1.94609 -0.57361 -1.87939  0.79216  1.78727 -1.00000 -1.67098  1.19432  1.53209 -1.37248 -2.00000  1.37248 -1.53209 -1.19432  1.67098  1.00000 -1.78727 -0.79216  1.87939  0.57361 -1.94609 -0.34730  1.98648  0.11629     ... ... ..... ....... ......... ........... .............
E8u  *  2.00000 -0.57361 -1.67098  1.53209  0.79216 -1.98648  0.34730  1.78727 -1.37248 -1.00000  1.94609 -0.11629 -1.87939  1.19432 -2.00000 -1.19432  1.87939  0.11629 -1.94609  1.00000  1.37248 -1.78727 -0.34730  1.98648 -0.79216 -1.53209  1.67098  0.57361     ... ... ..... ....... ......... ........... .............
E9u  *  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -1.00000  2.00000 -1.00000 -2.00000  1.00000 -2.00000  1.00000  1.00000 -2.00000  1.00000  1.00000 -2.00000  1.00000  1.00000 -2.00000  1.00000  1.00000     ... ... ..... ....... ......... ........... .............
E10u *  2.00000 -1.37248 -0.11629  1.53209 -1.98648  1.19432  0.34730 -1.67098  1.94609 -1.00000 -0.57361  1.78727 -1.87939  0.79216 -2.00000 -0.79216  1.87939 -1.78727  0.57361  1.00000 -1.94609  1.67098 -0.34730 -1.19432  1.98648 -1.53209  0.11629  1.37248     ... ... ..... ....... ......... ........... .............
E11u *  2.00000 -1.67098  0.79216  0.34730 -1.37248  1.94609 -1.87939  1.19432 -0.11629 -1.00000  1.78727 -1.98648  1.53209 -0.57361 -2.00000  0.57361 -1.53209  1.98648 -1.78727  1.00000  0.11629 -1.19432  1.87939 -1.94609  1.37248 -0.34730 -0.79216  1.67098     ... ... ..... ....... ......... ........... .............
E12u *  2.00000 -1.87939  1.53209 -1.00000  0.34730  0.34730 -1.00000  1.53209 -1.87939  2.00000 -1.87939  1.53209 -1.00000  0.34730 -2.00000 -0.34730  1.00000 -1.53209  1.87939 -2.00000  1.87939 -1.53209  1.00000 -0.34730 -0.34730  1.00000 -1.53209  1.87939     ... ... ..... ....... ......... ........... .............
E13u *  2.00000 -1.98648  1.94609 -1.87939  1.78727 -1.67098  1.53209 -1.37248  1.19432 -1.00000  0.79216 -0.57361  0.34730 -0.11629 -2.00000  0.11629 -0.34730  0.57361 -0.79216  1.00000 -1.19432  1.37248 -1.53209  1.67098 -1.78727  1.87939 -1.94609  1.98648     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.986476715484 = 2*cos(2*π/54) = 2*cos(π/27)
                               1.946089741160 = 2*cos(4*π/54) = 2*cos(2*π/27)
                               1.879385241572 = 2*cos(6*π/54) = 2*cos(π/9)
                               1.787265280647 = 2*cos(8*π/54) = 2*cos(4*π/27)
                               1.670975622826 = 2*cos(10*π/54) = 2*cos(5*π/27)
                               1.532088886238 = 2*cos(12*π/54) = 2*cos(2*π/9)
                               1.372483275737 = 2*cos(14*π/54) = 2*cos(7*π/27)
                               1.194317183406 = 2*cos(16*π/54) = 2*cos(8*π/27)
                               0.792159532078 = 2*cos(20*π/54) = 2*cos(10*π/27)
                               0.573606465422 = 2*cos(22*π/54) = 2*cos(11*π/27)
                               0.347296355334 = 2*cos(24*π/54) = 2*cos(4*π/9)
                               0.116289657821 = 2*cos(26*π/54) = 2*cos(13*π/27)



 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    {x2y2, xy}, {z2(x2y2), xyz2}, {z4(x2y2), xyz4} 
E3g  g+i+k+m      {xz(x2−3y2), yz(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4g  g+i+k+m      {(x2y2)2−4x2y2, xy(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E5g  i+k+m        {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6g  i+k+m        {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
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(x2y2), xyz}, {z3(x2y2), xyz3} 
E3u  f+h+j+l      {x(x2−3y2), y(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)} 
E4u  h+j+l        {z((x2y2)2−4x2y2), xyz(x2y2)} 
E5u  h+j+l        {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)} 
E6u  j+l 
E7u  j+l 
E8u  l 
E9u  l 

 Notes:

    α  The order of the S54 point group is 54, and the order of the principal axis (S54) is 54. The group has 28 irreducible representations.

    β  The S54 point group could also be named C27i, because the S54 axis is identical to a roto-inversion axis of order 27.

    γ  The S54 point group is isomorphic to C54 and C27h.

    δ  The S54 point group is generated by one single symmetry element, S54. Therefore, it is a cyclic group.

    ε  The lowest nonvanishing multipole moment in S54 is 4 (quadrupole moment).

    ζ  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The S54 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 52 cases have been combined into 26 two-dimensional representations that are no longer irreducible but have real-valued characters.
       Accordingly, 26 pairs of left and right rotations have been combined into one two-membered pseudo-class each.

    θ  The 26 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.

    κ  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.

This Character Table for the S54 point group was created by Gernot Katzer.

For other groups and some explanations, see the Main Page.