Character table for the C_5v point group

C5v     E       2 C5    2 C5^2  5 sv       <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1      1.0000  1.0000  1.0000  1.0000     ... ..T ....T ......T ........T T.........T ..T.........T
A2      1.0000  1.0000  1.0000 -1.0000     ..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   p+d+f+g+2h+2i+2j+2k+2l+3m     z, z2, z3, z4, x(x2−(5+2√5)y2)(x2−(5−2√5)y2), z5, xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), z6 
A2   R+h+i+j+k+l+2m                Rz, y((5+2√5)x2−y2)((5−2√5)x2−y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2) 
E1   R+p+d+f+2g+2h+3i+3j+3k+4l+4m  {Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {(x2−y2)2−4x2y2, xy(x2−y2)}, {xz3, yz3}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {xz4, yz4}, {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)}, {xz5, yz5} 
E2   d+2f+2g+2h+2i+3j+4k+4l+4m     {x2−y2, xy}, {x(x2−3y2), y(3x2−y2)}, {z(x2−y2), xyz}, {xz(x2−3y2), yz(3x2−y2)}, {z2(x2−y2), xyz2}, {xz2(x2−3y2), yz2(3x2−y2)}, {z3(x2−y2), xyz3}, {xz3(x2−3y2), yz3(3x2−y2)}, {z4(x2−y2), xyz4} 

 Notes:

    α  The order of the C5v point group is 10, and the order of the principal axis (C5) is 5. The group has 4 irreducible representations.

    β  The C5v point group is isomorphic to D5.

    γ  The C5v point group is generated by two symmetry elements, C5 and any σv.
       Also, the group may be generated from any two σv planes.

    δ  The group contains one set of symmetry planes σv intersecting in the principal (z) axis. The xz plane (but not the yz plane) is a member of that set.

    ε  The lowest nonvanishing multipole moment in C5v is 2 (dipole 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 just less than half the order of the principal axis.

    θ  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.

    ι  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.