# octic group

The *octic group* also known as the ${4}^{th}$ dihedral group^{}, is a non-Abelian group^{} with eight elements. It is traditionally denoted by ${D}_{4}$. This group is defined by the presentation^{}

$$ |

or, equivalently, defined by the multiplication table

$\cdot $ | $e$ | $s$ | ${s}^{2}$ | ${s}^{3}$ | $t$ | $ts$ | $t{s}^{2}$ | $t{s}^{3}$ |
---|---|---|---|---|---|---|---|---|

$e$ | $e$ | $s$ | ${s}^{2}$ | ${s}^{3}$ | $t$ | $ts$ | $t{s}^{2}$ | $t{s}^{3}$ |

$s$ | $s$ | ${s}^{2}$ | ${s}^{3}$ | $e$ | $t{s}^{3}$ | $t$ | $ts$ | $t{s}^{2}$ |

${s}^{2}$ | ${s}^{2}$ | ${s}^{3}$ | $e$ | $s$ | $t{s}^{2}$ | $t{s}^{3}$ | $t$ | $ts$ |

${s}^{3}$ | ${s}^{3}$ | $e$ | $s$ | ${s}^{2}$ | $ts$ | $t{s}^{2}$ | $t{s}^{3}$ | $t$ |

$t$ | $t$ | $ts$ | $t{s}^{2}$ | $t{s}^{3}$ | $e$ | $s$ | ${s}^{2}$ | ${s}^{3}$ |

$ts$ | $ts$ | $t{s}^{2}$ | $t{s}^{3}$ | $t$ | ${s}^{3}$ | $e$ | $s$ | ${s}^{2}$ |

$t{s}^{2}$ | $t{s}^{2}$ | $t{s}^{3}$ | $t$ | $ts$ | ${s}^{2}$ | ${s}^{3}$ | $e$ | $s$ |

$t{s}^{3}$ | $t{s}^{3}$ | $t$ | $ts$ | $t{s}^{2}$ | $s$ | ${s}^{2}$ | ${s}^{3}$ | $e$ |

where we have put each product^{} $xy$ into row $x$ and column $y$. The lattice of the subgroups^{} is given below:

where $$ denotes the subgroup generated by ${a}_{1},\mathrm{\dots},{a}_{n}$ and $\{{b}_{1},\mathrm{\dots},{b}_{n}\}$ denotes the subgroup. Of those subgroups, the following are its proper normal subgroup^{}: $\{e,{s}^{2},t,t{s}^{2}\}$, $$, $\{e,{s}^{2},st,ts\}$, and $$. In addition the center and commutator subgroup^{} of the octic group is $$. It can also be shown that the automorphism^{} of the octic group ($\mathrm{Aut}\mathit{}\mathrm{(}{D}_{\mathrm{4}}\mathrm{)}$) is isomorphic to itself(${D}_{\mathrm{4}}$).[PJ] An additional property is that the subgroup of the general linear group^{} of dimension^{} 2 over the real numbers generated by:
[
0
1
-1
0
],[
0
1
1
0
]
is isomorphic to the octic group.

## References

- PJ Pedersen, John: Groups of small order. http://www.math.usf.edu/ eclark/algctlg/small_groups.htmlhttp://www.math.usf.edu/ eclark/algctlg/small_groups.html

Title | octic group |
---|---|

Canonical name | OcticGroup |

Date of creation | 2013-03-22 14:47:59 |

Last modified on | 2013-03-22 14:47:59 |

Owner | Daume (40) |

Last modified by | Daume (40) |

Numerical id | 5 |

Author | Daume (40) |

Entry type | Example |

Classification | msc 20F55 |

Synonym | ${D}_{4}$ |