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Given a prime $p$, the group $UUT(3,p)$ is defined as follows: it is the [[unitriangular matrix group ]] of upper triangular matrices with 1s on the diagonal, and entries [[unitriangular matrix group of degree three|degree three]] over $F_p$ (with the group operation being matrix multiplication). Each such matrix $(a_{ij})$ can be described by the three entries [[prime field]] $a_\mathbb{12}, a_{13}, a_{23F}_p$. The matrix looks like: $\begin{pmatrix}1 & a_{12} & a_{13} \\0 & 1 & a_{23}\\0 & 0 & 1\end{pmatrix}$ The multiplication of matrices $A = (a_{ij})$ and $B = (b_{ij})$ gives the matrix <matH>C = (c_{ij})[/itex] where: * $c_{12} = a_{12} + b_{12}$* $c_{13} = a_{13} + b_{13} + a_{12}b_{23}$* $c_{23} = a_{23} + b_{23}$
The analysis given below does not apply to the case $p = 2$. For $p = 2$, we get the [[dihedral group:D8]], which is studied separately.