Supercharacter theories for dihedral group:D8

This page lists the various possible supercharacter theories for dihedral group:D8, i.e., the dihedral group with eight elements (this is the dihedral group of degree four and order eight). It builds on a thorough understanding of element structure of dihedral group:D8, subgroup structure of dihedral group:D8, and linear representation theory of dihedral group:D8.

We take $$D_8$$ to have the following presentation, with identity element $$e$$:

$$\langle a,x \mid a^4 = x^2 = e, xax = a^{-1} \rangle$$

Thus, $$a$$ is the generator of a cyclic maximal subgroup and $$x$$ is an element of order two outside that subgroup.

Character table
Below, the character table for $$D_8$$ is given.

Automorphism group subgroup actions
The outer automorphism group of dihedral group:D8 is cyclic group:Z2, with the non-identity element acting by interchanging the two conjugacy classes of elements of order two outside the cyclic maximal subgroup.

Galois group actions, or supercharacter theories based on character theories over subfields of the splitting field
The minimal splitting field in characteristic zero is $$\mathbb{Q}$$, so the only supercharacter theory we can get is the ordinary character theory.

Normal series
The summary table in the summary section lists all the supercharacter theories arising from nontrivial normal series. Note that the supercharacter theory where all non-identity elements form one block corresponds to the trivial normal series that includes only the trivial subgroup and the whole group.

Adjoint group structures
(the table below may be incomplete)