Cyclic group:Z18

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This group, denoted C_{18}, \Z_{18}, \Z/18\Z, is defined in the following equivalent ways:

  1. It is the direct product of cyclic group of order nine and cyclic group of order two.
  2. It is the cyclic group of order 18.

Arithmetic functions

Function Value Explanation
order 18
exponent 18
Frattini length 2

GAP implementation

Group ID

This finite group has order 18 and has ID 2 among the groups of order 18 in GAP's SmallGroup library. For context, there are groups of order 18. 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(18,2);

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

IdGroup(G) = [18,2]

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 defined using GAP's CyclicGroup function: