# Wreath product of Z4 and Z2

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]

## Contents

## Definition

This group is defined as the wreath product of the cyclic group of order four and the cyclic group of order two. In other words, it is the semidirect product where the acts by coordinate exchange.

## Position in classifications

Get more information about groups of the same order at Groups of order 32#The list

Type of classification | Position/number in classification |
---|---|

GAP ID | , i.e., among groups of order 32 |

Hall-Senior number | 31 among groups of order 32 |

Hall-Senior symbol |

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#Arithmetic functions

## Group properties

Want to compare and contrast group properties with other groups of the same order? Check out groups of order 32#Group properties

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

group of prime power order | Yes | ||

nilpotent group | Yes | prime power order implies nilpotent | |

supersolvable group | Yes | via nilpotent: finite nilpotent implies supersolvable | |

solvable group | Yes | via nilpotent: nilpotent implies solvable | |

abelian group | No |

## GAP implementation

### Group ID

This finite group has order 32 and has ID 11 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(32,11)`

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

`gap> G := SmallGroup(32,11);`

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

`IdGroup(G) = [32,11]`

or just do:

`IdGroup(G)`

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

### Other descriptions

The group can be described using GAP's WreathProduct and CyclicGroup functions:

`WreathProduct(CyclicGroup(4),CyclicGroup(2))`