# Wreath product of Z6 and Z2

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## Definition

This group is defined as the external wreath product with base cyclic group:Z6 and acting group cyclic group:Z2, where the latter acts via its regular group action, i.e., its action as permutations on a set of size tow.

More explicitly, it is the external semidirect product: $(\mathbb{Z}_6 \times \mathbb{Z}_6) \rtimes \mathbb{Z}_2$

where the non-identity element of the acting group $\mathbb{Z}_2$ acts by permuting the two copies of $\mathbb{Z}_6$, i.e., by a coordinate exchange automorphism.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 72#Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 72 groups with same order order of semidirect product is product of orders: the order is $6^2 \cdot 2$.

## GAP implementation

### Group ID

This finite group has order 72 and has ID 30 among the groups of order 72 in GAP's SmallGroup library. For context, there are 50 groups of order 72. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(72,30)

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

gap> G := SmallGroup(72,30);

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

IdGroup(G) = [72,30]

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(6),CyclicGroup(2)) WreathProduct, CyclicGroup