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)