In general, there are two classic ways to represent the <PERSON> group. The first one comes from its definition. They are the "rigid" motions for a 4D <PERSON> spacetime. Identifying space-time with $\mathbb R^4$ equipped with the metric: $$\eta = \begin{pmatrix} 1 & 0 & 0 & 0\ 0 & -1 & 0 & 0\ 0 & 0 & -1 & 0\ 0 & 0 & 0 & -1\ \end{pmatrix} $$ you can view the <PERSON> transformations as the affine transformation $f: x\in\mathbb R^4 \to\Lambda x+u$ with $u$ a constant fourvector and $\Lambda$ the analogue of a rotation matrix satisfying: $$ \Lambda \eta \Lambda^T = \eta $$ ie a linear transformation preserving the metric, more commonly known as a <PERSON> transformation.
Translations are just the case when $u=0$ and <PERSON> transformations about the origin when $\Lambda=I_4$. Within the <PERSON> transformations, you typically distinguish rotations and boosts. This is done by fixing a time coordinate (note that the distinction is observer dependent). 578
In our case, let's fix time to be the first coordinate $x^0$, with corresponding basis vector $e_0$.
Rotations are the subgroup of the <PERSON> group that fix $e_0$ ie don't mess with time. You can show that you can write: $$ \Lambda = \begin{pmatrix} 1 & 0 \ 0 & R \end{pmatrix} $$ with $R$ a rotation matrix ie a $3\times 3$ satisfying $RR^T = I_3$.
For boosts, you can't rigorously talk about a representation as they do not form a subgroup of the <PERSON> group (this is the origin of the famous <PERSON> precession). This subset can be parametrised by a $3$ vector $v$ with corresponding Lorentz factor $\gamma = (1-v^Tv)^{-1/2}$ by: $$ \Lambda = \begin{pmatrix} \gamma & \gamma v^T \ \gamma v & I_3+\frac{-1+\gamma}{v^T v}v v^T \end{pmatrix} $$
Note that there is another convenient way of representing the <PERSON> group as a matrix subgroup. It is the mathematical trick of homogenising The idea is to add an extra dimension. You can view them as $5\times 5$ the matrices of the form: $$ \begin{pmatrix} 1 & 0 \ u & \Lambda \end{pmatrix} $$ To prove the equivalence with the previous one, you can easily show it has the same action on the affine subspace $\begin{pmatrix}1 \ x\end{pmatrix}$ with $x\in\mathbb R^4$. In a similar way, you can view the Galilean transform more explicitly as a degenerate <PERSON> group by viewing the group of boosts and rotations as a matrix subgroup. In fact, this representation shows that it is in some sense a degenerate version of a bigger group, which leads to AdS and dS spaces.
Hope this helps 364