# Wreath product of Z3 and S3

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## Definition

This group is defined as the external wreath product of cyclic group:Z3 and symmetric group:S3, where the latter is taken as having its natural permutation action on a set of size three.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 162#Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 162 groups with same order order of semidirect product is product of orders: the order is $3^3 \cdot 3! = 27 \cdot 6 = 162$.

## GAP implementation

### Group ID

This finite group has order 162 and has ID 10 among the groups of order 162 in GAP's SmallGroup library. For context, there are groups of order 162. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(162,10)

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

gap> G := SmallGroup(162,10);

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

IdGroup(G) = [162,10]

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(3),SymmetricGroup(3)) WreathProduct, CyclicGroup, SymmetricGroup