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:

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

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 $m=5$: $4m=4(5)=20$
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 $m=5$: $2m=2(5)=10$.
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.

Subgroups

Item	Value
Number of subgroups	10
normal subgroups	5

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.