Direct product of A4 and Z2

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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
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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 \Z_2 \wr Z_3, or equivalently, the group (\Z_2 \times \Z_2 \times \Z_2) \rtimes \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 15 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))