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)