Character table for the D25h point group

D25h     E        2 C25    2 C25^2  2 C25^3  2 C25^4  2 C5     2 C25^6  2 C25^7  2 C25^8  2 C25^9  2 C5^2   2 C25^11 2 C25^12 25 C2'   sh       2 S25    2 S25^3  2 S5     2 S25^7  2 S25^9  2 S25^11 2 S25^13 2 S5^3   2 S25^17 2 S25^19 2 S25^21 2 S25^23 25 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     ... ... ....T ....... ........T ........... ............T
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     ... ... ..... ....... ......... ........... .............
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     ..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     ... ..T ..... ......T ......... ..........T .............
E1'     2.00000  1.93717  1.75261  1.45794  1.07165  0.61803  0.12558 -0.37476 -0.85156 -1.27485 -1.61803 -1.85955 -1.98423  0.00000  2.00000  1.93717  1.45794  0.61803 -0.37476 -1.27485 -1.85955 -1.98423 -1.61803 -0.85156  0.12558  1.07165  1.75261  0.00000     ... TT. ..... ....TT. ......... ........TT. .............
E1"     2.00000  1.93717  1.75261  1.45794  1.07165  0.61803  0.12558 -0.37476 -0.85156 -1.27485 -1.61803 -1.85955 -1.98423  0.00000 -2.00000 -1.93717 -1.45794 -0.61803  0.37476  1.27485  1.85955  1.98423  1.61803  0.85156 -0.12558 -1.07165 -1.75261  0.00000     TT. ... ..TT. ....... ......TT. ........... ..........TT.
E2'     2.00000  1.75261  1.07165  0.12558 -0.85156 -1.61803 -1.98423 -1.85955 -1.27485 -0.37476  0.61803  1.45794  1.93717  0.00000  2.00000  1.75261  0.12558 -1.61803 -1.85955 -0.37476  1.45794  1.93717  0.61803 -1.27485 -1.98423 -0.85156  1.07165  0.00000     ... ... TT... ....... ....TT... ........... ........TT...
E2"     2.00000  1.75261  1.07165  0.12558 -0.85156 -1.61803 -1.98423 -1.85955 -1.27485 -0.37476  0.61803  1.45794  1.93717  0.00000 -2.00000 -1.75261 -0.12558  1.61803  1.85955  0.37476 -1.45794 -1.93717 -0.61803  1.27485  1.98423  0.85156 -1.07165  0.00000     ... ... ..... ..TT... ......... ......TT... .............
E3'     2.00000  1.45794  0.12558 -1.27485 -1.98423 -1.61803 -0.37476  1.07165  1.93717  1.75261  0.61803 -0.85156 -1.85955  0.00000  2.00000  1.45794 -1.27485 -1.61803  1.07165  1.75261 -0.85156 -1.85955  0.61803  1.93717 -0.37476 -1.98423  0.12558  0.00000     ... ... ..... TT..... ......... ....TT..... .............
E3"     2.00000  1.45794  0.12558 -1.27485 -1.98423 -1.61803 -0.37476  1.07165  1.93717  1.75261  0.61803 -0.85156 -1.85955  0.00000 -2.00000 -1.45794  1.27485  1.61803 -1.07165 -1.75261  0.85156  1.85955 -0.61803 -1.93717  0.37476  1.98423 -0.12558  0.00000     ... ... ..... ....... ..TT..... ........... ......TT.....
E4'     2.00000  1.07165 -0.85156 -1.98423 -1.27485  0.61803  1.93717  1.45794 -0.37476 -1.85955 -1.61803  0.12558  1.75261  0.00000  2.00000  1.07165 -1.98423  0.61803  1.45794 -1.85955  0.12558  1.75261 -1.61803 -0.37476  1.93717 -1.27485 -0.85156  0.00000     ... ... ..... ....... TT....... ........... ....TT.......
E4"     2.00000  1.07165 -0.85156 -1.98423 -1.27485  0.61803  1.93717  1.45794 -0.37476 -1.85955 -1.61803  0.12558  1.75261  0.00000 -2.00000 -1.07165  1.98423 -0.61803 -1.45794  1.85955 -0.12558 -1.75261  1.61803  0.37476 -1.93717  1.27485  0.85156  0.00000     ... ... ..... ....... ......... ..TT....... .............
E5'     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  2.00000  0.61803 -1.61803  2.00000 -1.61803  0.61803  0.61803 -1.61803  2.00000 -1.61803  0.61803  0.61803 -1.61803  0.00000     ... ... ..... ....... ......... TT......... .............
E5"     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 -2.00000 -0.61803  1.61803 -2.00000  1.61803 -0.61803 -0.61803  1.61803 -2.00000  1.61803 -0.61803 -0.61803  1.61803  0.00000     ... ... ..... ....... ......... ........... ..TT.........
E6'     2.00000  0.12558 -1.98423 -0.37476  1.93717  0.61803 -1.85955 -0.85156  1.75261  1.07165 -1.61803 -1.27485  1.45794  0.00000  2.00000  0.12558 -0.37476  0.61803 -0.85156  1.07165 -1.27485  1.45794 -1.61803  1.75261 -1.85955  1.93717 -1.98423  0.00000     ... ... ..... ....... ......... ........... TT...........
E6"     2.00000  0.12558 -1.98423 -0.37476  1.93717  0.61803 -1.85955 -0.85156  1.75261  1.07165 -1.61803 -1.27485  1.45794  0.00000 -2.00000 -0.12558  0.37476 -0.61803  0.85156 -1.07165  1.27485 -1.45794  1.61803 -1.75261  1.85955 -1.93717  1.98423  0.00000     ... ... ..... ....... ......... ........... .............
E7'     2.00000 -0.37476 -1.85955  1.07165  1.45794 -1.61803 -0.85156  1.93717  0.12558 -1.98423  0.61803  1.75261 -1.27485  0.00000  2.00000 -0.37476  1.07165 -1.61803  1.93717 -1.98423  1.75261 -1.27485  0.61803  0.12558 -0.85156  1.45794 -1.85955  0.00000     ... ... ..... ....... ......... ........... .............
E7"     2.00000 -0.37476 -1.85955  1.07165  1.45794 -1.61803 -0.85156  1.93717  0.12558 -1.98423  0.61803  1.75261 -1.27485  0.00000 -2.00000  0.37476 -1.07165  1.61803 -1.93717  1.98423 -1.75261  1.27485 -0.61803 -0.12558  0.85156 -1.45794  1.85955  0.00000     ... ... ..... ....... ......... ........... .............
E8'     2.00000 -0.85156 -1.27485  1.93717 -0.37476 -1.61803  1.75261  0.12558 -1.85955  1.45794  0.61803 -1.98423  1.07165  0.00000  2.00000 -0.85156  1.93717 -1.61803  0.12558  1.45794 -1.98423  1.07165  0.61803 -1.85955  1.75261 -0.37476 -1.27485  0.00000     ... ... ..... ....... ......... ........... .............
E8"     2.00000 -0.85156 -1.27485  1.93717 -0.37476 -1.61803  1.75261  0.12558 -1.85955  1.45794  0.61803 -1.98423  1.07165  0.00000 -2.00000  0.85156 -1.93717  1.61803 -0.12558 -1.45794  1.98423 -1.07165 -0.61803  1.85955 -1.75261  0.37476  1.27485  0.00000     ... ... ..... ....... ......... ........... .............
E9'     2.00000 -1.27485 -0.37476  1.75261 -1.85955  0.61803  1.07165 -1.98423  1.45794  0.12558 -1.61803  1.93717 -0.85156  0.00000  2.00000 -1.27485  1.75261  0.61803 -1.98423  0.12558  1.93717 -0.85156 -1.61803  1.45794  1.07165 -1.85955 -0.37476  0.00000     ... ... ..... ....... ......... ........... .............
E9"     2.00000 -1.27485 -0.37476  1.75261 -1.85955  0.61803  1.07165 -1.98423  1.45794  0.12558 -1.61803  1.93717 -0.85156  0.00000 -2.00000  1.27485 -1.75261 -0.61803  1.98423 -0.12558 -1.93717  0.85156  1.61803 -1.45794 -1.07165  1.85955  0.37476  0.00000     ... ... ..... ....... ......... ........... .............
E10'    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  2.00000 -1.61803  0.61803  2.00000  0.61803 -1.61803 -1.61803  0.61803  2.00000  0.61803 -1.61803 -1.61803  0.61803  0.00000     ... ... ..... ....... ......... ........... .............
E10"    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 -2.00000  1.61803 -0.61803 -2.00000 -0.61803  1.61803  1.61803 -0.61803 -2.00000 -0.61803  1.61803  1.61803 -0.61803  0.00000     ... ... ..... ....... ......... ........... .............
E11'    2.00000 -1.85955  1.45794 -0.85156  0.12558  0.61803 -1.27485  1.75261 -1.98423  1.93717 -1.61803  1.07165 -0.37476  0.00000  2.00000 -1.85955 -0.85156  0.61803  1.75261  1.93717  1.07165 -0.37476 -1.61803 -1.98423 -1.27485  0.12558  1.45794  0.00000     ... ... ..... ....... ......... ........... .............
E11"    2.00000 -1.85955  1.45794 -0.85156  0.12558  0.61803 -1.27485  1.75261 -1.98423  1.93717 -1.61803  1.07165 -0.37476  0.00000 -2.00000  1.85955  0.85156 -0.61803 -1.75261 -1.93717 -1.07165  0.37476  1.61803  1.98423  1.27485 -0.12558 -1.45794  0.00000     ... ... ..... ....... ......... ........... .............
E12'    2.00000 -1.98423  1.93717 -1.85955  1.75261 -1.61803  1.45794 -1.27485  1.07165 -0.85156  0.61803 -0.37476  0.12558  0.00000  2.00000 -1.98423 -1.85955 -1.61803 -1.27485 -0.85156 -0.37476  0.12558  0.61803  1.07165  1.45794  1.75261  1.93717  0.00000     ... ... ..... ....... ......... ........... .............
E12"    2.00000 -1.98423  1.93717 -1.85955  1.75261 -1.61803  1.45794 -1.27485  1.07165 -0.85156  0.61803 -0.37476  0.12558  0.00000 -2.00000  1.98423  1.85955  1.61803  1.27485  0.85156  0.37476 -0.12558 -0.61803 -1.07165 -1.45794 -1.75261 -1.93717  0.00000     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.984229402629 = 2*cos(2*π/50) = 2*cos(π/25)
                               1.937166322257 = 2*cos(4*π/50) = 2*cos(2*π/25)
                               1.859552971777 = 2*cos(6*π/50) = 2*cos(3*π/25)
                               1.752613360088 = 2*cos(8*π/50) = 2*cos(4*π/25)
                               1.618033988750 = 2*cos(10*π/50) = 2*cos(π/5) = (√5+1)/2
                               1.457937254843 = 2*cos(12*π/50) = 2*cos(6*π/25)
                               1.274847979497 = 2*cos(14*π/50) = 2*cos(7*π/25)
                               1.071653589958 = 2*cos(16*π/50) = 2*cos(8*π/25)
                               0.851558583130 = 2*cos(18*π/50) = 2*cos(9*π/25)
                               0.618033988750 = 2*cos(20*π/50) = 2*cos(2*π/5) = (√5−1)/2
                               0.374762629171 = 2*cos(22*π/50) = 2*cos(11*π/25)
                               0.125581039059 = 2*cos(24*π/50) = 2*cos(12*π/25)



 Symmetry of Rotations and Cartesian products

A1'  d+g+i+k+m    z2, z4, z6 
A2'  R            Rz 
A2"  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        {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+m        {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E6"  j+l 
E7'  j+l 
E7"  k+m 
E8'  k+m 
E8"  l 
E9'  l 
E9"  m 
E10' m 

 Notes:

    α  The order of the D25h point group is 100, and the order of the principal axis (S25) is 50. The group has 28 irreducible representations.

    β  The D25h point group is isomorphic to D25d, C50v and D50.

    γ  The D25h point group is generated by two symmetry elements, S25 and either a perpendicular C2 or a vertical σv.
       Also, the group may be generated from two σv planes, or a σv and a C2 (some pairs will yield smaller groups, though; choosing a minimum angle is safe).
       The canonical choice, however, is to use redundant generators: C25, C2 and σh.

    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.
       Similarly, the single set of symmetry planes denoted σd contains the xz plane but not the yz plane.

    ε  The lowest nonvanishing multipole moment in D25h 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  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.

This Character Table for the D25h point group was created by Gernot Katzer.

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