Supercharacter theories for dihedral group:D8
This article gives specific information, namely, supercharacter theories, about a particular group, namely: dihedral group:D8.
View supercharacter theories for particular groups | View other specific information about 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 to have the following presentation, with identity element :
Thus, is the generator of a cyclic maximal subgroup and is an element of order two outside that subgroup.
Below, the character table for is given.
This character table works over characteristic zero:
|Representation/Conj class||(size 1)||(size 1)||(size 2)||(size 2)||(size 2)|
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.
|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 , so the only supercharacter theory we can get is the ordinary character theory.
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
(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|