# Dihedral group:D20

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## Definition

This group is defined in the following equivalent ways:

1. It is the dihedral group of order $20$.
2. It is the direct product of the dihedral group of order ten and the cyclic group of order two.

## Arithmetic functions

Function Value Explanation
order 20
exponent 10
Frattini length 1
Fitting length 1
derived length 2

## Group properties

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

## GAP implementation

### Group ID

This finite group has order 20 and has ID 4 among the groups of order 20 in GAP's SmallGroup library. For context, there are 5 groups of order 20. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(20,4)

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

gap> G := SmallGroup(20,4);

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

IdGroup(G) = [20,4]

or just do:

IdGroup(G)

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