Unitriangular matrix group:UT(3,4)

As a group of matrices
This group is the member of family::unitriangular matrix group of degree three (i.e., the group of $$3 \times 3$$ upper-triangular matrices with $$1$$s on the diagonal) over the field of four elements. It is isomorphic to the 2-Sylow subgroup of general linear group:GL(3,4), special linear group:SL(3,4), projective general linear group:PGL(3,4), projective special linear group:PSL(3,4).

The group can be described explicitly as:

$$\left \{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{13},a_{23} \in \mathbb{F}_4 \right \}$$

The multiplication of matrices $$A = (a_{ij})$$ and $$B = (b_{ij})$$ gives the matrix $$C = (c_{ij})$$ where:


 * $$c_{12} = a_{12} + b_{12}$$
 * $$c_{13} = a_{13} + b_{13} + a_{12}b_{23}$$
 * $$c_{23} = a_{23} + b_{23}$$

The identity element is the identity matrix.

The inverse of a matrix $$A = (a_{ij})$$ is the matrix $$M = (m_{ij})$$ where:


 * $$m_{12} = -a_{12}$$
 * $$m_{13} = -a_{13} + a_{12}a_{23}$$
 * $$m_{23} = -a_{23}$$

Note that all addition and multiplication in these definitions is happening over the field $$\mathbb{F}_4$$.