Dihedral group:D32

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is the dihedral group of degree sixteen and order thirty-two. It is given by the presentation:

$\langle a,x\mid a^{16}=x^{2}=e,xax=a^{-1}\rangle$.

Arithmetic functions

Function	Value	Explanation
order	32
exponent	16
derived length	2
nilpotency class	4
minimum size of generating set	2
subgroup rank	2

Group properties

Property	Satisfied	Explanation
abelian group	No
group of prime power order	Yes
nilpotent group	Yes
maximal class group	Yes
metacyclic group	Yes
metabelian group	Yes
maximal class group	Yes
directly indecomposable group	Yes
splitting-simple group	No
centrally indecomposable group	Yes

GAP implementation

Group ID

This finite group has order 32 and has ID 18 among the groups of order 32 in GAP's SmallGroup library. For context, there are groups of order 32. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(32,18)

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

gap> G := SmallGroup(32,18);

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

IdGroup(G) = [32,18]

or just do:

IdGroup(G)

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

Other descriptions

The group can be described using GAP's DihedralGroup function:

DihedralGroup(32)