# Difference between revisions of "Wreath product of Z3 and S3"

(Created page with "{{particular group}} ==Definition== This group is defined as the defining ingredient::external wreath product of defining ingredient::cyclic group:Z3 and [[defining ing...") |
(→Arithmetic functions) |
||

Line 8: | Line 8: | ||

{{compare and contrast arithmetic functions|order = 162}} | {{compare and contrast arithmetic functions|order = 162}} | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Function !! Value !! Similar groups !! Explanation | ||

+ | |- | ||

+ | | {{arithmetic function value order|162}} || [[order of semidirect product is product of orders]]: the order is <math>3^3 \cdot 3! = 27 \cdot 6 = 162</math>. | ||

+ | |} | ||

==GAP implementation== | ==GAP implementation== |

## Latest revision as of 19:41, 30 October 2011

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 external wreath product of cyclic group:Z3 and symmetric group:S3, where the latter is taken as having its natural permutation action on a set of size three.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 162#Arithmetic functions

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

order (number of elements, equivalently, cardinality or size of underlying set) | 162 | groups with same order | order of semidirect product is product of orders: the order is . |

## GAP implementation

### Group ID

This finite group has order 162 and has ID 10 among the groups of order 162 in GAP's SmallGroup library. For context, there are groups of order 162. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(162,10)`

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

`gap> G := SmallGroup(162,10);`

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

`IdGroup(G) = [162,10]`

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(3),SymmetricGroup(3)) |
WreathProduct, CyclicGroup, SymmetricGroup |