I have done a little bit of abstract algebra with groups. Most of it relates to rather practically oriented applications in engineering like rotation groups, permutations, affine transformations. For these purposes we have a tool known as representation theory, where a group representation is a mapping between every group element and a matrix, in such a way that matrix multiplication between these matrices correspond to the group operation:
$$ab=c \Leftrightarrow {\bf M_a M_b = M_c}, \forall a,b,c \in G$$
Furthermore I know there exists a concept of product between groups.
Now to the question, given two matrix representations with representation matrices $$\{\bf M_{1k}\},\{M_{2l}\}$$ for any two elements in their respective groups $$k\in G_1, l\in G_2$$
Can we construct a matrix representation for the product of these groups straight away from these $\{\bf M_{1k}\},\{\bf M_{2l}\}$ ?
My suspicion is that some kind of tensor product should allow us to do this, but I have no proof.
Yes, this is often called the outer tensor product of representations. If we state each representation as group homomorphisms $\phi: G \to \mathrm{GL}_n$ and $\psi: H \to \mathrm{GL}_m$, then we can construct a representation of $G \times H$ on the tensor product space $V = \mathbb{C}^n \otimes \mathbb{C}^m$ by setting the group element $(g, h)$ to act by $\phi(g) \otimes \psi(h)$. In terms of matrices, this is called the kronecker product of the matrices $\phi(g)$ and $\psi(h)$.