# Group of translations by a fixed distance

I think I need this group ( if it exists) to help solve a physics problem.
Not being a mathematician, what I'd really like is a matrix representation
and a rule for getting the covariant derivative in the event of a broken symmetry.

I'd be much obliged if any one can give me any information.

M.

$$\left(\begin{array}{ccccccc}1 && 0 && 0 && u \\ 0 && 1 && 0 && v\\ 0 && 0 && 1 && w \\ 0 && 0 && 0&& 1\end{array}\right)\left(\begin{array}{c}x \\ y \\ z \\ 1\end{array}\right)= \left(\begin{array}{c}x+ u \\ y+ v \\ z+ w \\ 1\end{array}\right)$$
$$\left(\begin{array}{ccccccc}1 && 0 && 0 && u \\ 0 && 1 && 0 && v\\ 0 && 0 && 1 && w \\ 0 && 0 && 0&& 1\end{array}\right)\left(\begin{array}{ccccccc}1 && 0 && 0 && u \\ 0 && 1 && 0 && v\\ 0 && 0 && 1 && w \\ 0 && 0 && 0&& 1\end{array}\right) = \left(\begin{array}{ccccccc}1 && 0 && 0 && 2u \\ 0 && 1 && 0 && 2v\\ 0 && 0 && 1 && 2w \\ 0 && 0 && 0&& 1\end{array}\right)$$