Direct product of D8 and Z5

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Definition

This group is defined as the external direct product of dihedral group:D8 and cyclic group:Z5.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 40#Arithmetic functions

Basic arithmetic functions

Function Value Similar groups Explanation for function value
order (number of elements, equivalently, cardinality or size of underlying set) 40 groups with same order order of direct product is product of orders
nilpotency class 2

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


GAP implementation

Group ID

This finite group has order 40 and has ID 10 among the groups of order 40 in GAP's SmallGroup library. For context, there are groups of order 40. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(40,10)

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

gap> G := SmallGroup(40,10);

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

IdGroup(G) = [40,10]

or just do:

IdGroup(G)

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


Other descriptions

Description Functions used
DirectProduct(DihedralGroup(8),CyclicGroup(5)) DirectProduct, DihedralGroup, CyclicGroup
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