# Unitriangular matrix group:UT(3,Z9)

## Definition

### As matrices

This group is defined as the unitriangular matrix group of degree three over ring:Z9. Explicitly, it is the group (under matrix multiplication) of upper-triangular $3 \times 3$ unipotent matrices over the ring $\mathbb{Z}/9\mathbb{Z}$, i.e., matrices of the form: $\left\{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} : a_{12}, a_{13}, a_{23} \in \mathbb{Z}/9\mathbb{Z} \right\}$

## GAP implementation

### Group ID

This finite group has order 729 and has ID 24 among the groups of order 729 in GAP's SmallGroup library. For context, there are groups of order 729. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(729,24)

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

gap> G := SmallGroup(729,24);

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

IdGroup(G) = [729,24]

or just do:

IdGroup(G)

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

### Description as a matrix group

This description uses the functions SL, ZmodnZ, IsZero, and IsOne.

gap> L := SL(3,ZmodnZ(4));;
gap> G := Group(Filtered(L,x -> ForAll([x,x,x],IsOne) and ForAll([x,x,x],IsZero)));;

### Description by presentation

gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^9,F.2^9,F.3^9,F.1*F.3*F.1^(-1)*F.3^(-1),F.1*F.2*F.1^(-1)*F.2^(-1),F.2*F.3*F.2^(-1)*F.3^(-1)*F.1^(-1)];
<fp group on the generators [ f1, f2, f3 ]>