# Nontrivial semidirect product of Z4 and Z4

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## Contents

## Definition

### A presentation as a metacyclic group

The group, a semidirect product , can be defined by:

.

## 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

## Group properties

Want to compare and contrast group properties with other groups of the same order? Check out groups of order 16#Group properties

Property | Satisfied? | Explanation | Comment |
---|---|---|---|

Abelian group | No | ||

Group of prime power order | Yes | ||

Nilpotent group | Yes | ||

Metabelian group | Yes | ||

Metacyclic group | Yes | ||

Supersolvable group | Yes | ||

Group of nilpotency class two | Yes | ||

T-group | No | ||

Directly indecomposable group | Yes | ||

Splitting-simple group | No | ||

UL-equivalent group | No | See also nilpotent not implies UL-equivalent |

## Subgroups

`Further information: subgroup structure of nontrivial semidirect product of Z4 and Z4`

In case a single equivalence class of subgroups under automorphisms comprises multiple conjugacy classes of subgroups, outer curly braces are used to bucket the conjugacy classes.

## GAP implementation

### Group ID

This finite group has order 16 and has ID 4 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(16,4)`

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

`gap> G := SmallGroup(16,4);`

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

`IdGroup(G) = [16,4]`

or just do:

`IdGroup(G)`

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

### Other descriptions

gap> G := F/[F.1^2, F.2^4, F.1*F.2*F.1^(-1)*F.2^(-1),F.3^4,F.3*F.1*F.3^(-1)*F.1^(-1),F.3*F.2*F.3^(-1)*F.2^(-1)*F.1^(-1), F.3^2 * F.2^2]; <fp group on the generators [ f1, f2, f3 ]> gap> IdGroup(G); [ 16, 4 ]