# Supercharacter theories for dihedral group:D8

## Contents |

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.

## Character table

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) |
---|---|---|---|---|---|

Trivial representation | 1 | 1 | 1 | 1 | 1 |

-kernel | 1 | 1 | 1 | -1 | -1 |

-kernel | 1 | 1 | -1 | 1 | -1 |

-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 , 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 |