Cyclic group:Z20

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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
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This group is defined in the following equivalent ways:

  1. It is the cyclic group of order 20.
  2. It is the direct product of the cyclic group of order five and cyclic group of order four.

Arithmetic functions

Function Value Explanation
order 20
exponent 20
Frattini length 2
Fitting length 1
derived length 1

Group properties

Property Satisfied Explanation
cyclic group Yes
abelian group Yes
metacyclic group Yes

GAP implementation

Group ID

This finite group has order 20 and has ID 2 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:


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

gap> G := SmallGroup(20,2);

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

IdGroup(G) = [20,2]

or just do:


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

Other descriptions

The group can be constructed using GAP's CyclicGroup function: