# Wreath product of Z2 and A4

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## Definition

This group is defined as the external wreath product of cyclic group:Z2 by alternating group:A4, where the permutation action of the latter is taken as the natural action on a set of size four.

More explicitly, it is the external semidirect product with base elementary abelian group:E16 (the direct product of four copies of cyclic group:Z2) by alternating group:A4: $(\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2) \rtimes A_4$

where the latter acts on the former by coordinate permutations given by the natural action of $A_4$ on a set of size four.

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 192 groups with same order order of semidirect product is product of orders. In this case, we get $2^4 \cdot 12$ where $2^4 = 16$ is the order of elementary abelian group:E16 and $12 = 4!/2$ is the order of alternating group:A4.
exponent of a group 12 groups with same order and exponent of a group | groups with same exponent of a group PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set

## GAP implementation

### Group ID

This finite group has order 192 and has ID 201 among the groups of order 192 in GAP's SmallGroup library. For context, there are 1,543 groups of order 192. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(192,201)

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

gap> G := SmallGroup(192,201);

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

IdGroup(G) = [192,201]

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(2),AlternatingGroup(4)) WreathProduct, CyclicGroup, AlternatingGroup