# Difference between revisions of "Direct product of A4 and S3"

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This group is defined as the [[defining ingredient::external direct product]] of [[defining ingredient::alternating group:A4]] and [[defining ingredient::symmetric group:S3]]. | This group is defined as the [[defining ingredient::external direct product]] of [[defining ingredient::alternating group:A4]] and [[defining ingredient::symmetric group:S3]]. | ||

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+ | ==Arithmetic functions== | ||

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+ | {{compare and contrast arithmetic functions|order = 72}} | ||

+ | |||

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

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

+ | |- | ||

+ | | {{arithmetic function value order|72}} || [[order of direct product is product of orders]]: the order is <math>12 \times 6</math>, where <math>12 = 4!/2</math> is the order of [[alternating group:A4]] and <math>6 = 3!</math> is the order of [[symmetric group:S3]]. | ||

+ | |- | ||

+ | | {{arithmetic function value given order|exponent of a group|6|72}} || [[exponent of direct product is lcm of exponents]]: the exponent is <math>\operatorname{lcm} \{ 6,6 \} = 6</math>. | ||

+ | |- | ||

+ | | {{arithmetic function value given order|derived length|2|72}} || [[derived length of direct product is maximum of derived lengths]]: the derived length is <math>\max \{ 2,2 \} = 2</math>. | ||

+ | |} | ||

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

## Latest revision as of 03:46, 29 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 direct product of alternating group:A4 and symmetric group:S3.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 72#Arithmetic functions

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

order (number of elements, equivalently, cardinality or size of underlying set) | 72 | groups with same order | order of direct product is product of orders: the order is , where is the order of alternating group:A4 and is the order of symmetric group:S3. |

exponent of a group | 6 | groups with same order and exponent of a group | groups with same exponent of a group | exponent of direct product is lcm of exponents: the exponent is . |

derived length | 2 | groups with same order and derived length | groups with same derived length | derived length of direct product is maximum of derived lengths: the derived length is . |

## GAP implementation

### Group ID

This finite group has order 72 and has ID 44 among the groups of order 72 in GAP's SmallGroup library. For context, there are 50 groups of order 72. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(72,44)`

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

`gap> G := SmallGroup(72,44);`

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

`IdGroup(G) = [72,44]`

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 |
---|---|

DirectProduct(AlternatingGroup(4),SymmetricGroup(3)) |
DirectProduct, AlternatingGroup, SymmetricGroup |