The dihedral group of order 8

Example

The dihedral group $D_4$ 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 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 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 $S_4$.