Direct product of Z16 and V4

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]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is defined as the uses as intermediate construct:external direct product of the cyclic group of order 16 and Klein four-group (i.e., the elementary abelian group of order four). In other words, it is given by the presentation:

$G : = ⟨ a, b, c ∣ a^{16} = b^{2} = c^{2} = e, a b = b a, b c = c b, a c = c a ⟩$ .

It is the abelian group of prime power order corresponding to the prime $2$ and the partition $6 = 4 + 1 + 1$ .

Arithmetic functions

Function	Value	Explanation
order	64
exponent	16
nilpotency class	1
derived length	1
Frattini length	4
Fitting length	1
minimum size of generating set	3
subgroup rank	3
rank as p-group	3
normal rank as p-group	3
characteristic rank as p-group	3

Group properties

Property	Satisfied?	Explanation
cyclic group	No
homocyclic group	No
metacyclic group	No
elementary abelian group	No
abelian group	Yes
nilpotent group	Yes
solvable group	Yes

GAP implementation

Group ID

This finite group has order 64 and has ID 183 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(64,183)

For instance, we can use the following assignment in GAP to create the group and name it $G$ :

gap> G := SmallGroup(64,183);

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

IdGroup(G) = [64,183]

or just do:

IdGroup(G)

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

Other descriptions

The group can be defined using GAP's DirectProduct, CyclicGroup, and ElementaryAbelianGroup as:

DirectProduct(CyclicGroup(16),CyclicGroup(2),CyclicGroup(2))

or

DirectProduct(CyclicGroup(16),ElementaryAbelianGroup(4))