# Character table for the D5 point group

D5      E       2 C5    2 C5^2  5 C2'      <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————>
A1      1.0000  1.0000  1.0000  1.0000     ... ... ....T ....... ........T T.......... ...T........T
A2      1.0000  1.0000  1.0000 -1.0000     ..T ..T ..... ......T ......... .T........T ..T..........
E1      2.0000  0.6180 -1.6180  0.0000     TT. TT. ..TT. ....TT. TT....TT. ..TT....TT. TT..TT....TT.
E2      2.0000 -1.6180  0.6180  0.0000     ... ... TT... TTTT... ..TTTT... ....TTTT... ......TTTT...

Irrational character values:  0.618033988750 = 2*cos(2*π/5) = (√5−1)/2
-1.618033988750 = 2*cos(4*π/5) = −(√5+1)/2

Symmetry of Rotations and Cartesian products

A1   d+g+h+2i+j+2k+l+3m            z2, z4, x(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2), z6
A2   R+p+f+2h+i+2j+k+2l+2m         Rz, z, z3, y((5+2√5)x2y2)((5−2√5)x2y2), z5, xz(x2−(5+2√5)y2)(x2−(5−2√5)y2)
E1   R+p+d+f+2g+2h+3i+3j+3k+4l+4m  {Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {(x2y2)2−4x2y2, xy(x2y2)}, {xz3, yz3}, {z((x2y2)2−4x2y2), xyz(x2y2)}, {xz4, yz4}, {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)}, {xz5, yz5}
E2   d+2f+2g+2h+2i+3j+4k+4l+4m     {x2y2, xy}, {x(x2−3y2), y(3x2y2)}, {z(x2y2), xyz}, {xz(x2−3y2), yz(3x2y2)}, {z2(x2y2), xyz2}, {xz2(x2−3y2), yz2(3x2y2)}, {z3(x2y2), xyz3}, {xz3(x2−3y2), yz3(3x2y2)}, {z4(x2y2), xyz4}

Notes:

α  The order of the D5 point group is 10, and the order of the principal axis (C5) is 5. The group has 4 irreducible representations.

β  The D5 point group is isomorphic to C5v.

γ  The D5 point group is generated by two symmetry elements, C5 and a perpendicular C2.
Also, the group may be generated from any two C2 axes.

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.

ε  The lowest nonvanishing multipole moment in D5 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  The point group is chiral, as it does not contain any mirroring operation.

θ  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 just less than half the order of the principal axis.

ι  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.

κ  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.

This Character Table for the D5 point group was created by Gernot Katzer.

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