# Algebraic group interpretations of dihedral group:D8

From Groupprops

This article gives specific information, namely, algebraic group interpretations, about a particular group, namely: dihedral group:D8.

View algebraic group interpretations of particular groups | View other specific information about dihedral group:D8

## Unitriangular matrix group interpretation

### Description

The group dihedral group:D8 can be interpreted as the group of -points of a unipotent algebraic group over the algebraic closure of , namely the unitriangular matrix group of degree three . Explicitly, , which in turn is the set of -fixed points of .

This is the only possible structure for as the -fixed points of a unipotent algebraic group.

In fact, since the group scheme corresponding to the unitriangular matrix group of degree three is integral (i.e., makes sense over ), analogues of the group can be defined over any ring.

### Field extensions

Field extension of field:F2 | Degree of extension | Size of field | Size of group = cube of size of field | Group of points over that field in |
---|---|---|---|---|

field:F4 | 2 | 4 | 64 | unitriangular matrix group:UT(3,4) |

field:F8 | 3 | 8 | 512 | unitriangular matrix group:UT(3,8) |

### Local rings with this as residue field

Local ring | Length of ring | Size of ring | Size of group = cube of size of ring | Corresponding algebraic group type notion |
---|---|---|---|---|

ring:Z4 | 2 | 4 | 64 | unitriangular matrix group:UT(3,Z4) |

2 | 4 | 64 | unitriangular matrix group of degree three over quotient of polynomial ring over F2 by square of indeterminate | |

ring:Z8 | 3 | 8 | 512 | unitriangular matrix group:UT(3,Z8) |