Unitriangular matrix group:UT(3,7)

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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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Definition

This group is defined in the following equivalent ways:

  1. It is the unitriangular matrix group of degree three over field:F7. In other words, it is the group UT(3,p) for p = 7.
  2. It is the unique non-abelian group of order 7^3 and exponent 7.
  3. It is the extraspecial group of order 7^3.

GAP implementation

Group ID

This finite group has order 343 and has ID 3 among the groups of order 343 in GAP's SmallGroup library. For context, there are groups of order 343. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(343,3)

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

gap> G := SmallGroup(343,3);

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

IdGroup(G) = [343,3]

or just do:

IdGroup(G)

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