# Wreath product of Z2 and A5

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## Definition

This group is defined as the external wreath product of cyclic group:Z2 by alternating group:A5, where the permutation action of the latter is taken to be its natural action on a set of size five.

More explicitly, it is the external semidirect product of elementary abelian group:E32 by alternating group:A5: $(\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2) \rtimes A_5$

where $A_5$ acts on the group $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ (which is elementary abelian group:E32) by permuting the coordinates based on its natural action on a set of size five.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 1920#Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 1920 groups with same order The order is $2^5 \cdot 60$ (see order of semidirect product is product of orders). Here $2^5 = 32$ is the order of the base of the semidirect product and $60 = 5!/2$ is the order of the acting group, alternating group:A5.
exponent of a group 60 groups with same order and exponent of a group | groups with same exponent of a group
derived length -- -- not a solvable group
nilpotency class -- -- not a nilpotent group
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set Denote the identity and non-identity elements of $\mathbb{Z}_2$ by 0 and 1 and the elements of $A_5$ by permutations in cycle decomposition notation. Then, every element of the semidirect product is an ordered pair of a 5-tuple from $\mathbb{Z}_2$ and an element of $A_5$. The elements $((1,0,0,0,0),(3,4,5))$ and $((0,0,0,0,0),(1,2,3,4,5))$ generate the whole group.

## GAP implementation

### Group ID

This finite group has order 1920 and has ID 240997 among the groups of order 1920 in GAP's SmallGroup library. For context, there are 241,004 groups of order 1920. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(1920,240997)

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

gap> G := SmallGroup(1920,240997);

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

IdGroup(G) = [1920,240997]

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