# Direct product of Z8 and Z4 and V4

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## Contents

## Definition

This group is a finite abelian group of order given as the direct product of cyclic group:Z8, cyclic group:Z4, and Klein four-group.

## As an abelian group of prime power order

This group is the abelian group of prime power order corresponding to the unordered integer partition:

and the prime . In other words, it is the group .

## Note

This group is the smallest example of a finite abelian group within which we can find series-equivalent subgroups that are not automorphic subgroups. For more, see series-equivalent not implies automorphic in finite abelian group.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 128#Arithmetic functions

## GAP implementation

### Group ID

This finite group has order 128 and has ID 1601 among the groups of order 128 in GAP's SmallGroup library. For context, there are groups of order 128. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(128,1601)`

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

`gap> G := SmallGroup(128,1601);`

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

`IdGroup(G) = [128,1601]`

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 |
---|---|

DirectProduct(CyclicGroup(8),CyclicGroup(4),CyclicGroup(2),CyclicGroup(2)) |
DirectProduct, CyclicGroup |