Supercharacter theories for dihedral group:D8

This article gives specific information, namely, supercharacter theories, about a particular group, namely: dihedral group:D8.
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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.

This character table works over characteristic zero:

Representation/Conj class	$\{e \}$ (size 1)	$\{ a^2 \}$ (size 1)	$\{ a, a^{-1} \}$ (size 2)	$\{ x, a^2x \}$ (size 2)	$\{ ax, a^3x \}$ (size 2)
Trivial representation	1	1	1	1	1
$\langle a \rangle$-kernel	1	1	1	-1	-1
$\langle a^2, x \rangle$-kernel	1	1	-1	1	-1
$\langle a^2, ax\rangle$-kernel	1	1	-1	-1	1
2-dimensional	2	-2	0	0	0

The same character table works over any characteristic not equal to 2 where the elements 1,-1,0,2,-2 are interpreted over the field.

Supercharacter theories

Summary

Quick description of supercharacter theory	Number of such supercharacter theories under automorphism group action	Number of blocks of conjugacy classes = number of blocks of irreducible representations	Block sizes for conjugacy classses (in number of conjugacy class terms) (should add up to 5, the total number of conjugacy classes)	Block sizes for conjugacy classes (in number of elements terms) (should add up to 8, the order of the group)	Block sizes for irreducible representations (in number of representations terms) (should add up to 5, the total number of conjugacy classes)	Block sizes for irreducible representations (in sum of squares of degrees terms) (should add up to 8, the order of the group)
ordinary character theory	1	5	1,1,1,1,1	1,1,2,2,2	1,1,1,1,1	1,1,1,1,4
all non-identity elements form one block	1	2	1,4	1,7	1,4	1,7
supercharacter theory corresponding to normal series with middle group center of dihedral group:D8	1	3	1,1,3	1,1,6	1,3,1	1,3,4
supercharacter theory corresponding to normal series with middle group cyclic maximal subgroup of dihedral group:D8	1	3	1,2,2	1,3,4	1,1,3	1,1,6
supercharacter theory corresponding to normal series with middle group one of the Klein four-subgroups of dihedral group:D8	2	3	1,2,2	1,3,4	1,1,3	1,1,6
superchararacter theory corresponding to normal series that goes through center of dihedral group:D8 and cyclic maximal subgroup of dihedral group:D8 (note: this is also the supercharacter theory corresponding to the action of the whole automorphism group)	1	4	1,1,1,2	1,1,2,4	1,1,2,1	1,1,2,4
superchararacter theory corresponding to normal series that goes through center of dihedral group:D8 and one of the Klein four-subgroups of dihedral group:D8	2	4	1,1,1,2	1,1,2,4	1,1,2,1	1,1,2,4
Total (7 rows)	9 (number of supercharacter theories)	--	--	--	--	--

Sources of supercharacter theories

Automorphism group subgroup actions

Further information: endomorphism structure of dihedral group:D8

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.

Subgroup of outer automorphism group	Corresponding supercharacter theory
trivial subgroup	ordinary character theory
whole group	superchararacter theory corresponding to normal series that goes through center of dihedral group:D8 and cyclic maximal subgroup of dihedral group:D8

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

Further information: supercharacter theory corresponding to a 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

Further information: algebra group structures for dihedral group:D8, adjoint group structures for dihedral group:D8, supercharacter theory of the adjoint group of a radical ring

(the table below may be incomplete)

Adjoint group structure	Corresponding supercharacter theory in terms of explicit description of superconjugacy classes
algebra group structure over field:F2 corresponding to unitriangular matrices	ordinary character theory