# Dihedral group:D12

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## Contents

## Definition

This group, usually denoted (though denoted in an alternate convention) is defined in the following equivalent ways:

- It is the dihedral group of order twelve. In other words, it is the dihedral group of degree six, i.e., the group of symmetries of a regular hexagon.
- It is the direct product of the symmetric group of degree three and the cyclic group of order two.
- It is the outer linear group of degree two over the field of two elements, i.e., the group .
- It is Borel subgroup of general linear group for general linear group:GL(2,3), i.e., the general linear group of degree two over field:F3.

The usual presentation is:

.

With this presentation, the symmetric group of degree three is the direct factor and the complement of order two is the subgroup .

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 12#Arithmetic functions

Function | Value | Explanation |
---|---|---|

order | 12 | |

exponent | 6 | |

nilpotency class | -- | not a nilpotent group. |

derived length | 2 | |

Frattini length | 1 | |

Fitting length | 2 | |

minimum size of generating set | 2 | |

subgroup rank | 2 | |

max-length | 3 |

## GAP implementation

### Group ID

This finite group has order 12 and has ID 4 among the groups of order 12 in GAP's SmallGroup library. For context, there are 5 groups of order 12. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(12,4)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(12,4);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [12,4]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Other definitions

Description | Functions used |
---|---|

DihedralGroup(12) |
DihedralGroup |

DirectProduct(SymmetricGroup(3),CyclicGroup(2)) |
DirectProduct, SymmetricGroup, CyclicGroup |