Dihedral group:D20

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
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 defined in the following equivalent ways:

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

Definition by presentation

The dihedral group $D_{20}$ , sometimes called $D_{10}$ , also called the dihedral group of order twenty or the dihedral group of degree ten (since its natural action is on ten elements) is defined by the following presentation, with $e$ denoting the identity element:

$\langle x,a\mid a^{10}=x^{2}=e,xax^{-1}=a^{-1}\rangle$

Here, the element $a$ is termed the rotation or the generator of the cyclic piece and $x$ is termed the reflection.

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

Subgroups

Item	Value
Number of subgroups	22
normal subgroups	7

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 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.