# Character table for the S8 point group

```S8      E       2 S8    2 C4    2 S8^3  C2         <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————>
A       1.0000  1.0000  1.0000  1.0000  1.0000     ..T ... ....T ....... ........T ..TT....... ............T
B       1.0000 -1.0000  1.0000 -1.0000  1.0000     ... ..T ..... ......T TT....... ..........T ....TT.......
E1  *   2.0000  1.4142  0.0000 -1.4142 -2.0000     ... TT. ..... ....TT. ..TT..... ........TT. ..TT..TT.....
E2  *   2.0000  0.0000 -2.0000  0.0000  2.0000     ... ... TT... ..TT... ....TT... ......TT... TT......TT...
E3  *   2.0000 -1.4142  0.0000  1.4142 -2.0000     TT. ... ..TT. TT..... ......TT. TT..TT..... ..........TT.

Irrational character values:  1.414213562373 = 2*cos(2*π/8) = 2*cos(π/4) = √2

Symmetry of Rotations and Cartesian products

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

Notes:

α  The order of the S8 point group is 8, and the order of the principal axis (S8) is 8. The group has 5 irreducible representations.

β  The S8 point group could also be named C8i, because the S8 axis is identical to a roto-inversion axis of order 8.

γ  The S8 point group is isomorphic to C8.

δ  The S8 point group is generated by one single symmetry element, S8. Therefore, it is a cyclic group.

ε  The lowest nonvanishing multipole moment in S8 is 4 (quadrupole moment).

ζ  This is an Abelian point group (the commutative law holds between all symmetry operations).
The S8 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 6 cases have been combined into 3 two-dimensional representations that are no longer irreducible but have real-valued characters.
Accordingly, 3 pairs of left and right rotations have been combined into one two-membered pseudo-class each.

θ  The 3 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  Character tables of S8 published in various textbooks erroneously put the rotations (Rx,Ry) into the E1 representation, instead of E3.
See Randall B. Shirts, J. Chem. Educ., 2007, 84 (11), p 1882
```

This Character Table for the S8 point group was created by Gernot Katzer.

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