# Direct product of Z4 and V4

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

This group can be defined in a number of equivalent ways:

- It is the external direct product of a cyclic group of order four and a Klein four-group.
- It is the external direct product of a cyclic group of order four, and two copies of a cyclic group of order two.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

## GAP implementation

### Group ID

This finite group has order 16 and has ID 10 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(16,10)`

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

`gap> G := SmallGroup(16,10);`

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

`IdGroup(G) = [16,10]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Alternative descriptions

Description | Functions used |
---|---|

DirectProduct(CyclicGroup(4),ElementaryAbelianGroup(4)) |
DirectProduct, CyclicGroup, ElementaryAbelianGroup |

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