# Wreath product of Z2 and A5

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## Contents

## Definition

This group is defined as the external wreath product of cyclic group:Z2 by alternating group:A5, where the permutation action of the latter is taken to be its natural action on a set of size five.

More explicitly, it is the external semidirect product of elementary abelian group:E32 by alternating group:A5:

where acts on the group (which is elementary abelian group:E32) by permuting the coordinates based on its natural action on a set of size five.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 1920#Arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 1920 | groups with same order | The order is (see order of semidirect product is product of orders). Here is the order of the base of the semidirect product and is the order of the acting group, alternating group:A5. |

exponent of a group | 60 | groups with same order and exponent of a group | groups with same exponent of a group | |

derived length | -- | -- | not a solvable group |

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

minimum size of generating set | 2 | groups with same order and minimum size of generating set | groups with same minimum size of generating set | Denote the identity and non-identity elements of by 0 and 1 and the elements of by permutations in cycle decomposition notation. Then, every element of the semidirect product is an ordered pair of a 5-tuple from and an element of . The elements and generate the whole group. |

## GAP implementation

### Group ID

This finite group has order 1920 and has ID 240997 among the groups of order 1920 in GAP's SmallGroup library. For context, there are 241,004 groups of order 1920. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(1920,240997)`

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

`gap> G := SmallGroup(1920,240997);`

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

`IdGroup(G) = [1920,240997]`

or just do:

`IdGroup(G)`

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

### Other descriptions

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

WreathProduct(CyclicGroup(2),AlternatingGroup(5)) |
WreathProduct, CyclicGroup, AlternatingGroup |