Direct product of E4 and Z9

From Groupprops
Jump to: navigation, search
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]


This group is defined in the following equivalent ways:

  1. It is the direct product of the Klein four-group and the cyclic group of order nine.
  2. It is the direct product of the cyclic group of order eighteen and the cyclic group of order two.

Arithmetic functions

Function Value Explanation
order 36
exponent 18

GAP implementation

Group ID

This finite group has order 36 and has ID 5 among the groups of order 36 in GAP's SmallGroup library. For context, there are groups of order 36. 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(36,5);

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

IdGroup(G) = [36,5]

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 DirectProduct, ElementaryAbelianGroup, and CyclicGroup functions: