# Difference between revisions of "Semidirect product of Z3 and D8 with action kernel V4"

(Created page with '{{particular group}} ==Definition== This group is defined by the following presentation: <math>G := \langle a_1, a_2,a_3,a_4 \mid a_1^3 = a_2^2 = a_3^2 = a_4^2 = e, a_1a_2...') |
(→Arithmetic functions) |
||

(2 intermediate revisions by the same user not shown) | |||

Line 3: | Line 3: | ||

==Definition== | ==Definition== | ||

− | This [[group]] is defined by the | + | This [[group]] is defined as follow: it is the [[external semidirect product]] of [[cyclic group:Z3]] by [[dihedral group:D8]] where the action of the latter on the former is given by a homomorphism <math>D_8 \to \operatorname{Aut}(\mathbb{Z}_3)</math> whose kernel is one of the [[Klein four-subgroups of dihedral group:D8]]. Note that the homomorphism is completely specified by its kernel because there is a unique isomorphism between any two groups isomorphic to [[cyclic group:Z2]]. |

− | + | ==Arithmetic functions== | |

+ | |||

+ | {{compare and contrast arithmetic functions|order = 24}} | ||

+ | |||

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

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

+ | |- | ||

+ | | {{arithmetic function value order|24}} || [[order of semidirect product is product of orders]]: <math>|\mathbb{Z}_3||D_8| = 3 \cdot 8 = 24</math> | ||

+ | |- | ||

+ | | {{arithmetic function value given order|exponent of a group|12|24}} || [[exponent of a finite group equals product of exponents of its Sylow subgroups]]: In this case, by construction, the Sylow subgroups are <math>D_8</math> and <math>\mathbb{Z}_3</math>, so the exponent is the product of their exponent, which is <math>4 \times 3 = 12</math>. | ||

+ | |- | ||

+ | | [[nilpotency class]] || not a nilpotent group || || | ||

+ | |- | ||

+ | | {{arithmetic function value given order|derived length|2|24}} || The derived subgroup of the acting group <math>D_8</math> is in the kernel of the action and is abelian, and so is the base group <math>\mathbb{Z}_3</math>. | ||

+ | |- | ||

+ | | {{arithmetic function value given order|Fitting length|2|24}} || | ||

+ | |- | ||

+ | | {{arithmetic function value given order|Frattini length|2|24}} || | ||

+ | |- | ||

+ | | {{arithmetic function value given order|minimum size of generating set|2|24}} || | ||

+ | |} | ||

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

{{GAP ID|24|8}} | {{GAP ID|24|8}} | ||

− | |||

− | |||

− | |||

− | |||

− | |||

− |

## Latest revision as of 19:11, 9 June 2012

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]

## Definition

This group is defined as follow: it is the external semidirect product of cyclic group:Z3 by dihedral group:D8 where the action of the latter on the former is given by a homomorphism whose kernel is one of the Klein four-subgroups of dihedral group:D8. Note that the homomorphism is completely specified by its kernel because there is a unique isomorphism between any two groups isomorphic to cyclic group:Z2.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 24#Arithmetic functions

## GAP implementation

### Group ID

This finite group has order 24 and has ID 8 among the groups of order 24 in GAP's SmallGroup library. For context, there are 15 groups of order 24. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(24,8)`

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

`gap> G := SmallGroup(24,8);`

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

`IdGroup(G) = [24,8]`

or just do:

`IdGroup(G)`

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