Suppose I have a group $G$ with a presentation of the following form
$$G = \langle g_1, ... g_n \mid g_1^{a_1} = ... = g_n^{a_n} = [g_1,g_2]^{b_{12}} = ... = [g_{n-1},g_n]^{b_{n{-}1,n}} = e \rangle$$
is it possible to calculate the order of $G$ from the integers $a_i$ and $b_{ij}$? Or to see if $G$ is finite or infinite from looking at $a_i$ and $b_{ij}$?
is this the most general possible form for a group presentation? (That is, is it possible to write all conditions like $(g_1g_2)^2 = e$ or $(g_1g_2g_3)^4 = e$ etc. In terms of powers of commutators of two elements of the group?) Is there some way to generalize this presentation to give a presentation for a general finite group?
For context; I'm coming from a background in mathematical physics where we learn about Lie group theory, but I'm only just learning about discrete groups. I'm trying to see if there's any way that it makes sense to port over my intuition from Lie group theory; for example the generators in the group presentation to me look reminiscent of a basis for the Lie algebra, and then the conditions in the presentation look something like a prescription for what happens to the algebra under exponentiation. Whether the group is finite or infinite looks like if the manifold of a Lie group is compact or not. Maybe these intuitive (to me) analogies are just wrong, I don't know. Any comments on this or related reading would be interesting for me.