# Algebra group structures for dihedral group:D8

From Groupprops

This article gives specific information, namely, algebra group structures, about a particular group, namely: dihedral group:D8.

View algebra group structures for particular groups | View other specific information about dihedral group:D8

The group dihedral group:D8 has at least one (and probably only one?) algebra group structure over field:F2. It does not have any algebra group structure over any other fields.

## Algebra

### Multiplication table (structure constants)

The algebra is a three-dimensional algebra. We can describe it by means of the following multiplication table in terms of structure constants . The multiplication table is as follows:

0 | 0 | ||

0 | 0 | 0 | |

0 | 0 | 0 |

### Verification of properties

- The algebra is associative: All products of length three or more are zero.
- The algebra is nilpotent: All products of length three or more are zero.
- The algebra group is isomorphic to dihedral group:D8: is the central element of order 2, is the order four generator of the cyclic maximal subgroup. and are reflections outside this subgroup.

### Description as subalgebra of niltriangular matrix Lie algebra

The algebra is the *whole* of niltriangular matrix Lie algebra:NT(3,2), so dihedral group:D8 is isomorphic to . The explicit description is: