Direct product of A4 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 group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is defined in the following equivalent ways:

1. it is the external direct product of the alternating group of degree four and the cyclic group of order two.
2. It is the wreath product of the cyclic group of order two and the cyclic group of order three, acting regularly. In other words, it is the group $\mathbb {Z} _{2}\wr Z_{3}$, or equivalently, the group $(\mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2})\rtimes \mathbb {Z} _{3}$, where the latter acts on the former by cylic permutations of coordinates.

Arithmetic functions

Function	Value	Explanation
order	24
exponent	12
derived length	2
nilpotency class	--	not a nilpotent group.
Frattini length	1
minimum size of generating set	2
subgroup rank	3

Group properties

Property	Satisfied	Explanation
cyclic group	No
abelian group	No
nilpotent group	No
metacyclic group	No
metabelian group	Yes
supersolvable group	No
solvable group	Yes

Subgroup-defining functions

Subgroup-defining function	Subgroup type in list	Isomorphism class	Comment
center		cyclic group:Z2
commutator subgroup		Klein four-group
Frattini subgroup		trivial group
Fitting subgroup		elementary abelian group:E8

GAP implementation

Group ID

This finite group has order 24 and has ID 13 among the groups of order 24 in GAP's SmallGroup library. For context, there are groups of order 24. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(24,13)

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

gap> G := SmallGroup(24,13);

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

IdGroup(G) = [24,13]

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 defined using GAP's WreathProduct and CyclicGroup functions:

WreathProduct(CyclicGroup(2),CyclicGroup(3))

The group can be defined using GAP's DirectProduct, AlternatingGroup and CyclicGroup functions:

DirectProduct(AlternatingGroup(4),CyclicGroup(2))