Decompose a group $G$ into a normal subgroup $N$ and quotient group $Q$

Question

1. If $G$ has a normal subgroup $N$ and quotient group $Q$, so $$G/N=Q,$$ or $$N \to G \to Q$$

   can we always decompose a group $G$ into a pair of data $(N,Q)$? Say $g \in G$, $g$ is isomorphic to $(n,q)$ with $n \in N$ and $q \in Q$?

2. How can we generic express $g_1 \cdot g_2 \in G$, in terms of $$(n_1,q_1) \cdot (n_2,q_2)=(n', q') \in G?$$

How do we write explicitly $$(n', q')$$ in terms of $n_1,q_1,n_2,q_2$ as directly as possible?

We may assume a finite group for simplicity. (A side question: How about a Lie group?)