Dicyclic group:Dic20

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Definition

This group is defined as the dicyclic group of order $20$, and hence degree $5$. In other words, it has the presentation:

$\langle a,b,c\mid a^{5}=b^{2}=c^{2}=abc\rangle$

Alternatively, it has the presentation:

$\langle a,b,c\mid a^{10}=b^{4}=e,bab^{-1}=a^{-1}\rangle$.

Arithmetic functions

Function	Value	Explanation
order	20
exponent	10
Frattini length	2
derived length	2
nilpotency class	--	Not a nilpotent group.
minimum size of generating set	2
subgroup rank	2

Group properties

Property	Satisfied	Explanation
cyclic group	No
abelian group	No
nilpotent group	No
metacyclic group	Yes
supersolvable group	Yes
solvable group	Yes
ambivalent group	No

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 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 $G$:

gap> G := SmallGroup(20,1);

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

IdGroup(G) = [20,1]

or just do:

IdGroup(G)

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