# Difference between revisions of "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:

$\langle a,b,c \mid a^{10} = b^4 = e, a^5 = b^2, bab^{-1} = a^{-1} \rangle$.

## 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 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 $G$:

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.