Character table for the C15h point group

C15h    E        2 C15    2 C15^2  2 C5     2 C15^4  2 C3     2 C5^2   2 C15^7  sh       2 S15    2 S5     2 S3     2 S15^7  2 S5^3   2 S15^11 2 S15^13    <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A'      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
A"      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 .............
E1' *   2.00000  1.82709  1.33826  0.61803 -0.20906 -1.00000 -1.61803 -1.95630  2.00000  1.82709  0.61803 -1.00000 -1.95630 -1.61803 -0.20906  1.33826     ... TT. ..... ....TT. ......... ........TT. .............
E1" *   2.00000  1.82709  1.33826  0.61803 -0.20906 -1.00000 -1.61803 -1.95630 -2.00000 -1.82709 -0.61803  1.00000  1.95630  1.61803  0.20906 -1.33826     TT. ... ..TT. ....... ......TT. ........... ..........TT.
E2' *   2.00000  1.33826 -0.20906 -1.61803 -1.95630 -1.00000  0.61803  1.82709  2.00000  1.33826 -1.61803 -1.00000  1.82709  0.61803 -1.95630 -0.20906     ... ... TT... ....... ....TT... ........... ........TT...
E2" *   2.00000  1.33826 -0.20906 -1.61803 -1.95630 -1.00000  0.61803  1.82709 -2.00000 -1.33826  1.61803  1.00000 -1.82709 -0.61803  1.95630  0.20906     ... ... ..... ..TT... ......... ......TT... .............
E3' *   2.00000  0.61803 -1.61803 -1.61803  0.61803  2.00000  0.61803 -1.61803  2.00000  0.61803 -1.61803  2.00000 -1.61803  0.61803  0.61803 -1.61803     ... ... ..... TT..... ......... ....TT..... .............
E3" *   2.00000  0.61803 -1.61803 -1.61803  0.61803  2.00000  0.61803 -1.61803 -2.00000 -0.61803  1.61803 -2.00000  1.61803 -0.61803 -0.61803  1.61803     ... ... ..... ....... ..TT..... ........... ......TT.....
E4' *   2.00000 -0.20906 -1.95630  0.61803  1.82709 -1.00000 -1.61803  1.33826  2.00000 -0.20906  0.61803 -1.00000  1.33826 -1.61803  1.82709 -1.95630     ... ... ..... ....... TT....... ........... ....TT.......
E4" *   2.00000 -0.20906 -1.95630  0.61803  1.82709 -1.00000 -1.61803  1.33826 -2.00000  0.20906 -0.61803  1.00000 -1.33826  1.61803 -1.82709  1.95630     ... ... ..... ....... ......... ..TT....... .............
E5' *   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     ... ... ..... ....... ......... TT......... .............
E5" *   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     ... ... ..... ....... ......... ........... ..TT.........
E6' *   2.00000 -1.61803  0.61803  0.61803 -1.61803  2.00000 -1.61803  0.61803  2.00000 -1.61803  0.61803  2.00000  0.61803 -1.61803 -1.61803  0.61803     ... ... ..... ....... ......... ........... TT...........
E6" *   2.00000 -1.61803  0.61803  0.61803 -1.61803  2.00000 -1.61803  0.61803 -2.00000  1.61803 -0.61803 -2.00000 -0.61803  1.61803  1.61803 -0.61803     ... ... ..... ....... ......... ........... .............
E7' *   2.00000 -1.95630  1.82709 -1.61803  1.33826 -1.00000  0.61803 -0.20906  2.00000 -1.95630 -1.61803 -1.00000 -0.20906  0.61803  1.33826  1.82709     ... ... ..... ....... ......... ........... .............
E7" *   2.00000 -1.95630  1.82709 -1.61803  1.33826 -1.00000  0.61803 -0.20906 -2.00000  1.95630  1.61803  1.00000  0.20906 -0.61803 -1.33826 -1.82709     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.956295201468 = 2*cos(2*π/30) = 2*cos(π/15) = (√5−1+√30+6*√5)/4
                               1.827090915285 = 2*cos(4*π/30) = 2*cos(2*π/15) = (1+√5+√30−6*√5)/4
                               1.618033988750 = 2*cos(6*π/30) = 2*cos(π/5) = (√5+1)/2
                               1.338261212718 = 2*cos(8*π/30) = 2*cos(4*π/15) = (1−√5+√30+6*√5)/4
                               0.618033988750 = 2*cos(12*π/30) = 2*cos(2*π/5) = (√5−1)/2
                               0.209056926535 = 2*cos(14*π/30) = 2*cos(7*π/15) = (√30−6*√5−√5−1)/4



 Symmetry of Rotations and Cartesian products

A'   R+d+g+i+k+m  Rz, z2, z4, z6 
A"   p+f+h+j+l    z, z3, z5 
E1'  p+f+h+j+l    {x, y}, {xz2, yz2}, {xz4, yz4} 
E1"  R+d+g+i+k+m  {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} 
E2'  d+g+i+k+m    {x2y2, xy}, {z2(x2y2), xyz2}, {z4(x2y2), xyz4} 
E2"  f+h+j+l      {z(x2y2), xyz}, {z3(x2y2), xyz3} 
E3'  f+h+j+l      {x(x2−3y2), y(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)} 
E3"  g+i+k+m      {xz(x2−3y2), yz(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4'  g+i+k+m      {(x2y2)2−4x2y2, xy(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E4"  h+j+l        {z((x2y2)2−4x2y2), xyz(x2y2)} 
E5'  h+j+l+m      {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)} 
E5"  i+k+m        {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6'  i+k+l+m      {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E6"  j+l+m 
E7'  j+k+l+m 
E7"  k+l+m 

 Notes:

    α  The order of the C15h point group is 30, and the order of the principal axis (S15) is 30. The group has 16 irreducible representations.

    β  The C15h point group could also be named S15, as it contains the S15 axis as its only symmetry element.
       Another rare designation is C30i because the S15 axis is identical to a roto-inversion axis of order 30.

    γ  The C15h point group is isomorphic to C30 and S30.

    δ  The C15h point group is generated by one single symmetry element, S15. Therefore, it is a cyclic group.
       The canonical choice, however, is to use redundant generators: C15 and σh.

    ε  The lowest nonvanishing multipole moment in C15h is 4 (quadrupole moment).

    ζ  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The C15h 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 28 cases have been combined into 14 two-dimensional representations that are no longer irreducible but have real-valued characters.
       Accordingly, 14 pairs of left and right rotations have been combined into one two-membered pseudo-class each.

    θ  The 14 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 constructible polygon, as the order of the principal axis is a product of any number
       of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an
       algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers.

    λ  The fact that the regular pentagon is constructible is known since antiquity; Eukleides already discovered a construction for it.
       The double cosine of 2π/5 is equal to the reciprocal of the Golden Ratio of (1+√5)/2 = 1.61803.
       The regular 15-gon can be derived from the regular pentagon by overlaying it with a equilateral triangle.

This Character Table for the C15h point group was created by Gernot Katzer.

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