Dihedral group:D10
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]
Contents
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 :
gap> G := SmallGroup(10,1);
Conversely, to check whether a given group 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)