# Wreath product of Z5 and Z2

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## Definition

This group is defined as the external wreath product of cyclic group:Z5 and cyclic group:Z2, where the latter acts via the regular group action on a set of size two. In other words, it is an external semidirect product:

$(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_2$

where the latter acts on the former 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 50#Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 50 groups with same order order of semidirect product is product of orders: The order is $5^2 \cdot 2$
exponent of a group 10 groups with same order and exponent of a group | groups with same exponent of a group exponent of semidirect product is a multiple of lcm of exponents, exponent of extension group divides product of exponents of normal subgroup and quotient group. Both numbers are 10 in this case, so the exponent is 10.

## GAP implementation

### Group ID

This finite group has order 50 and has ID 3 among the groups of order 50 in GAP's SmallGroup library. For context, there are groups of order 50. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(50,3)

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

gap> G := SmallGroup(50,3);

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

IdGroup(G) = [50,3]

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(5),CyclicGroup(2)) WreathProduct, CyclicGroup