# Dihedral group:D10

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## Definition

This group is defined as the dihedral group of order ten. In other words, it is the semidirect product of the cyclic group of order five and a cyclic group of order two.

## Arithmetic functions

Function Value Explanation
order 10
exponent 10
Fitting length 2
Frattini length 1
derived length 2

## Group properties

Property Satisfied Explanation
cyclic group No
abelian group No
nilpotent group No
metacyclic group Yes
supersolvable group Yes
solvable group Yes

## GAP implementation

### Group ID

This finite group has order 10 and has ID 1 among the groups of order 10 in GAP's SmallGroup library. For context, there are groups of order 10. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(10,1)

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

gap> G := SmallGroup(10,1);

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

IdGroup(G) = [10,1]

or just do:

IdGroup(G)

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

### Other descriptions

This group can be defined using GAP's DihedralGroup function:

DihedralGroup(10)