# Wreath product of A4 and Z2

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## Definition

This group is defined as the external wreath product of alternating group:A4 and cyclic group:Z2, where the latter acts via the regular group action. More explicitly it is the external semidirect product:

$(A_4 \times A_4) \rtimes \mathbb{Z}_2$

where the non-identity element of $\mathbb{Z}_2$ acts by the coordinate exchange automorphism on $A_4 \times A_4$.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 288#Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 288 groups with same order order of semidirect product is product of orders: order is $12^2 \cdot 2 = 288$, where $12 = 4!/2$ is the order of alternating group:A4.

## GAP implementation

### Group ID

This finite group has order 288 and has ID 1025 among the groups of order 288 in GAP's SmallGroup library. For context, there are 1,045 groups of order 288. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(288,1025)

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

gap> G := SmallGroup(288,1025);

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

IdGroup(G) = [288,1025]

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