# Unitriangular matrix group:UT(3,7)

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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.