Direct product of Z3 and SmallGroup(40,8)

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Definition

This group is defined as:

  1. It is the external direct product of cyclic group:Z3 and SmallGroup(40,8).

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 80 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 20 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(120,20)

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

gap> G := SmallGroup(120,20);

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

IdGroup(G) = [120,20]

or just do:

IdGroup(G)

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


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