# Difference between revisions of "Direct product of D8 and Z2"

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Under the action of the automorphism group, the conjugacy classes (3) and (4) are in the same orbit -- in other words, <math>y</math> and <math>a^2y</math> are related by an automorphism. The conjugacy classes (5) and (6) are equivalent, and the conjugacy classes (7)-(10) are equivalent. Thus, the equivalence classes have sizes <math>1,1,2,4,8</math>. | Under the action of the automorphism group, the conjugacy classes (3) and (4) are in the same orbit -- in other words, <math>y</math> and <math>a^2y</math> are related by an automorphism. The conjugacy classes (5) and (6) are equivalent, and the conjugacy classes (7)-(10) are equivalent. Thus, the equivalence classes have sizes <math>1,1,2,4,8</math>. | ||

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+ | ==Arithmetic functions== | ||

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## Revision as of 22:42, 16 August 2009

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

*This particular group is a finite group of order:* 16

## Contents

## Definition

This group is defined as the external direct product of the dihedral group of order eight and the cyclic group of order two.

A presentation for it is:

.

## Elements

### Upto conjugacy

There are ten conjugacy classes:

- The identity element. (1)
- The element . (1)
- The element . (1)
- The element . (1)
- The two-element conjugacy class comprising and . (2)
- The two-element conjugacy class comprising and . (2)
- The two-element conjugacy class comprising and . (2)
- The two-element conjugacy class comprising and . (2)
- The two-element conjugacy class comprising and . (2)
- The two-element conjugacy class comprising and . (2)

### Upto automorphism

Under the action of the automorphism group, the conjugacy classes (3) and (4) are in the same orbit -- in other words, and are related by an automorphism. The conjugacy classes (5) and (6) are equivalent, and the conjugacy classes (7)-(10) are equivalent. Thus, the equivalence classes have sizes .

## Arithmetic functions

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

order | 16 | |

exponent | 4 | Cyclic subgroup of order four in dihedral part. |

nilpotency class | 2 | |

derived length | 2 | |

Frattini length | 2 | |

Fitting length | 1 | |

minimum size of generating set | 3 | Generating set of dihedral direct factor and generating element for other direct factor (also equal to rank of Frattini quotient). |

subgroup rank | 3 | |

max-length | 4 | |

rank as p-group | 3 | Direct product of Klein four-subgroup of first direct factor with second direct factor. |

normal rank | 3 | Direct product of Klein four-subgroup of first direct factor with second direct factor. |

characteristic rank of a p-group | 2 | Center is a Klein four-subgroup, no characteristic elementary abelian subgroups of order eight |

## Group properties

Property | Satisfied | Explanation | Comment |
---|---|---|---|

Abelian group | No | and don't commute | |

Nilpotent group | Yes | Prime power order implies nilpotent | |

Metacyclic group | No | ||

Supersolvable group | Yes | Finite nilpotent implies supersolvable | |

Solvable group | Yes | Nilpotent implies solvable | |

T-group | No | , which is normal, but is not normal | Smallest example for normality is not transitive. |

Monolithic group | No | Center is Klein four-group, any subgroup of order two is minimal normal. | |

One-headed group | No | Distinct maximal subgroups of order eight. | |

SC-group | No | ||

ACIC-group | Yes | Every automorph-conjugate subgroup is characteristic | |

Rational group | Yes | Any two elements that generate the same cyclic group are conjugate | Thus, all characters are integer-valued. |

Rational-representation group | Yes | All representations over characteristic zero are realized over the rationals. | Contrast with quaternion group, that is rational but not rational-representation. |

## Subgroups

`Further information: Subgroup structure of direct product of D8 and Z2`

The group has the following subgroups:

- The trivial group. (1)
- The cyclic group of order two. This is central, and is also the set of squares. Isomorphic to cyclic group:Z2. (1)
- The subgroups and . These are both central subgroups of order two, but are related by an outer automorphism. Isomorphic to cyclic group:Z2. (2)
- The subgroups , , , , , , , and . These are all related by automorphisms, and are all 2-subnormal subgroups. They come in four conjugacy classes, namely the class , the class , the class , and the class . Isomorphic to cyclic group:Z2. (8)
- The subgroup . This is the center, hence is a characteristic subgroup. Isomorphic to Klein four-group. (1)
- The subgroups , , , and . These are all normal subgroups, but are related by outer automorphisms. Isomorphic to Klein four-group. (4)
- The subgroups , , , , , , , . These subgroups are all 2-subnormal subgroups and are related by outer automorphisms, and they come in four conjugacy classes of size two. Isomorphic to Klein four-group. (8)
- The subgroups and . They are both normal and are related via an outer automorphism. Isomorphic to cyclic group:Z4. (2)
- The subgroups and . These are normal and are related by outer automorphisms. Isomorphic to elementary abelian group of order eight. (2)
- The subgroups , , and . These are both normal and are related by an outer automorphism. Isomorphic to dihedral group:D8. (4)
- The subgroup . This is a characteristic subgroup. Isomorphic to direct product of Z4 and Z2. (1)
- The whole group. (1)

## GAP implementation

### Group ID

The ID of this group in GAP's list of groups of order is . The group can be described using GAP's SmallGroup function as:

SmallGroup(16,11)

### Other descriptions

It can be described as a direct product:

DirectProduct(DihedralGroup(8),CyclicGroup(2))