Direct product of Z16 and Z4

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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 can be defined as the external direct product of the cyclic group of order sixteen and the cyclic group of order four. Alternatively, it has the presentation:

G := \langle a,b \mid a^{16} = b^4 = e, ab = ba \rangle.

GAP implementation

Group ID

This finite group has order 64 and has ID 26 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. 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(64,26);

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

IdGroup(G) = [64,26]

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 and DirectProduct functions as: