# Dicyclic group:Dic20

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
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]

## 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.

## 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.