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# Dihedral group:D8

### From Groupprops

*This article is about a particular group, viz a group unique upto isomorphism*

*This particular group is the smallest (in terms of order):* non-T-group

*This particular group is the smallest (in terms of order):* nilpotent non-Abelian group

*This particular group is a finite group of order:* 8

## Contents |

## Definition

### Definition by presentation

The **dihedral group** *D*_{8}, sometimes called *D*_{4}, also called the dihedral group of order eight or the dihedral group acting on four elements, is defined by the following presentation:

### Geometric definition

The **dihedral group** *D*_{4} is defined as the group of all symmetries of the square (the regular 4-gon). This has a cyclic subgroup comprising rotations (which is the cyclic subgroup generated by *a*) and has four *reflections* each being an involution: reflections about lines joining midpoints of opposite sides, and reflections about diagonals.

### Multiplication table

## Elements

### Upto conjugation

There are five conjugacy classes of elements of the dihedral group:

- The identity element
- The rotation by π, which is given as
*a*^{2}in the presentation - The two-element conjugacy class comprising rotations by , namely
*a*and*a*^{3} - The two-element conjugacy class comprising the two reflections:
*x*,*x**a*^{2} - The two-element conjugacy class comprising the two reflections:
*x**a*,*x**a*^{3}

### Upto automorphism

Under the equivalence relation of automorphisms, the last two conjugacy classes merge into one. There are thus four equivalence classes under the actions of automorphisms, of sizes 1, 1, 2 and 4.

## Subgroup-defining functions

### Center

*The center of this group is abstractly isomorphic to*: cyclic group of order two

The center of the quaternion group is the two-element subgroup comprising the identity and *a*^{2} (rotation by π)

### Commutator subgroup

*The commutator subgroup of this group is abstractly isomorphic to:* cyclic group of order two

The commutator subgroup of the dihedral group is the same as its center.

In particular this shows that the dihedral group is a group of nilpotence class two.

### Frattini subgroup

*The Frattini subgroup of this group is abstractly isomorphic to:* cyclic group of order two

The Frattini subgroup coincides with the center and commutator subgroup. This dihedral group is thus an extraspecial group.

### Socle

*The socle of this group is abstractly isomorphic to:* cyclic group of order two

The center is the unique minimal normal subgroup, and hence is also the socle.