Direct product of Z5 and SmallGroup(24,1)

From Groupprops

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:

  1. It is the external direct product of cyclic group:Z5 and SmallGroup(24,1).

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 120#Arithmetic functions

Basic arithmetic functions

Function Value Similar groups Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set) 120 groups with same order order of direct product is product of orders
nilpotency class not a nilpotent group

Arithmetic functions of a counting nature

Function Value Similar groups Explanation for function value
number of subgroups 20 groups with same order and number of subgroups | groups with same number of subgroups

Group properties

Property Satisfied? Explanation
abelian group No
nilpotent group No

GAP implementation

Group ID

This finite group has order 120 and has ID 1 among the groups of order 120 in GAP's SmallGroup library. For context, there are groups of order 120. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(120,1)

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

gap> G := SmallGroup(120,1);

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

IdGroup(G) = [120,1]

or just do:

IdGroup(G)

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


Retrieved from "https://groupprops.subwiki.org/w/index.php?title=Direct_product_of_Z5_and_SmallGroup(24,1)&oldid=54137"