Dicyclic group:Dic20
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Contents
Definition
This group is defined as the dicyclic group of order , and hence degree
. In other words, it has the presentation:
Alternatively, it has the presentation:
.
Arithmetic functions
Function | Value | Similar groups | Explanation |
---|---|---|---|
order (number of elements, equivalently, cardinality or size of underlying set) | 20 | groups with same order | As dicyclic group of degree ![]() ![]() |
exponent of a group | 10 | groups with same order and exponent of a group | groups with same exponent of a group | As dicyclic group of degree ![]() ![]() |
Frattini length | 2 | groups with same order and Frattini length | groups with same Frattini length | The Frattini subgroup is isomorphic to cyclic group:Z2 -- specifically, it is the center. |
derived length | 2 | groups with same order and derived length | groups with same derived length | The group is in fact a metacyclic group, hence metabelian, but it is not abelian. |
nilpotency class | -- | -- | Not a nilpotent group. |
minimum size of generating set | 2 | groups with same order and minimum size of generating set | groups with same minimum size of generating set | |
subgroup rank of a group | 2 | groups with same order and subgroup rank of a group | groups with same subgroup rank of a group |
Group properties
Important properties
Property | Satisfied? | Explanation |
---|---|---|
cyclic group | No | |
abelian group | No | |
nilpotent group | No | |
metacyclic group | Yes | |
supersolvable group | Yes | |
solvable group | Yes |
Other properties
Property | Satisfied? | Explanation |
---|---|---|
ambivalent group | No | |
Schur-trivial group | Yes | This is true for all dicyclic groups. |
finite group with periodic cohomology | Yes | This is true for all dicyclic groups. |
GAP implementation
Group ID
This finite group has order 20 and has ID 1 among the groups of order 20 in GAP's SmallGroup library. For context, there are 5 groups of order 20. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(20,1)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(20,1);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [20,1]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.