# Wreath product of Z2 and Z4

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## Definition

This group is defined as the wreath product of the cyclic group of order two and the cyclic group of order four with the regular group action. In other words, it is the semidirect product $(\Z_2 \times \Z_2 \times \Z_2 \times \Z_2) \rtimes \Z_4$, where the acting $\Z_4$ acts by cyclic permutation of coordinates.

## GAP implementation

### Group ID

This finite group has order 64 and has ID 32 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(64,32)

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

gap> G := SmallGroup(64,32);

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

IdGroup(G) = [64,32]

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 GAP:CyclicGroup functions:

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