If we want to "extract the group away from its representation", then we need to fully know the group structure, i.e. how group elements relate to each other via group multiplication. One way to do this is to build the multiplication table of the group, which explicitly shows all possible multiplications between elements (in other words it defines the group). However, continuous group or more specifically Lie Groups (such as Poincare group) have infinite elements and therefore such multiplication table is out of the question. Yet we want to be able to describe the group structure without mentioning any representation. What we do is to look at the group generators and its Lie algebra.
The algebra satisfied by the generators of the group consists in a set of algebraic relations independent of representation and which almost fully define the group. The way we obtain the generators is by looking at infinitesimal group transformation. By definition, Lie groups are differentiable manifolds thus we can find infinitesimal transformations by Taylor expanding around the identity element and we define the generator $T_a$ associated to the parameter $\theta_a$ as
$$g(\delta\theta_a)=e+i\delta\theta_a T_a,\quad\mathrm{no\, sum},$$
where $e$ is the identity. I said that the Lie Algebra (or equivalently the local structure of a Lie Group) almost fully determine the group because the same Lie Algebra is in general associated to more than one group. For example, both groups $SU(2)$ and $SO(3)$ have the same algebra $\mathfrak{su}(2)$. In order to associate the algebra with a particular group we also need to specify the representation, which is the same for algebra and group.
