---
tag: 054SBCQ93T
kind: example
name: The dihedral group of order 8
msc: [20B25, 20D15]
instance_of: [0JZ27DMJ5D]
uses: [09GK3DE0WV, 0S7Y9Q990Q]
sources:
  - url: https://en.wikipedia.org/wiki/Dihedral_group_of_order_8
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Dihedral_group_of_order_8
    locator: Symmetries of a square
  - url: https://en.wikipedia.org/wiki/Dihedral_group_of_order_8
    locator: Normal subgroups
same_as:
  wikidata: Q115556877
  wikipedia: Dihedral group of order 8
---

## Example

The [[dihedral group|0JZ27DMJ5D]] $D_4$ is the group of the symmetries of a square, and it has order $8$.
Let $r$ be the rotation by $90$ degrees and let $s$ be a reflection in a line of symmetry.
Then $D_4 = \{e, r, r^2, r^3, s, rs, r^2s, r^3s\}$, with $r^4 = e$, $s^2 = e$ and $srs^{-1} = r^{-1}$.
The elements $r$ and $r^3$ have order $4$, and the other five elements different from $e$ have order $2$.
The [[centre|0DZ93493VN]] of $D_4$ is $\{e, r^2\}$, because the rotation by $180$ degrees commutes with each symmetry.
The rotation subgroup $\langle r \rangle$ is [[normal|0CB0DX5BVV]] of index $2$.
The subgroup $\{e, s\}$ is not normal, because $r s r^{-1} = r^2 s$.
Numbering the vertices $1, 2, 3, 4$ gives an injective homomorphism from $D_4$ into $S_4$, and its image is a Sylow $2$-subgroup of $S_4$.