# Direct product of D8 and Z2

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

## Contents

## Definition

This group is defined in the following ways:

- It is the external direct product of the dihedral group of order eight and the cyclic group of order two.
- It is the generalized dihedral group corresponding to the direct product of Z4 and Z2.
- It is the -Sylow subgroup of the symmetric group of degree six.

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

## 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. |

Directly indecomposable group | No | ||

Centrally indecomposable group | No | ||

UL-equivalent group | No | It is a direct product of groups of different nilpotency class values | See also nilpotent not implies UL-equivalent |

## 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 all normal and are related via outer automorphisms. 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

This finite group has order 16 and has ID 11 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(16,11)`

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

`gap> G := SmallGroup(16,11);`

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

`IdGroup(G) = [16,11]`

or just do:

`IdGroup(G)`

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

### Other descriptions

It can be described as a direct product:

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