Recently, I was reading about Janko's discovery of $J_1$, the first “modern” sporadic simple group. Janko and others were trying to classify all finite simple groups with an involution centralizer isomorphic to $C_2 \times \mathrm{PSL}(2,q)$. At some point in that process, it seemed possible that there exist a new sporadic simple group with involution centralizer $C_2 \times \mathrm{PSL}(2,5)$ among them. People (mostly Janko himself) started to investigate this hypothetical simple group $J_1$. They computed its order (175560) and drew conclusions about its subgroup structure. In the end, Janko was able to give concrete generators of $J_1$ as a subgroup of $\mathrm{GL}(7,11)$ (the matrices can be found on Wikipedia). At this point it was known that if that new sporadic group $J_1$ really exists, then it has to be a concrete subgroup of $\mathrm{GL}(7,11)$. Afterwards, the existence of $J_1$ was confirmed by a computer program.
Can someone offer more insights about how it was possible to get from some structural knowledge of this hypothetical group to concrete generators as a matrix group? I am curious to see how to derive a linear action out of a hypothetical simple group (of course one cannot just simply pick an elementary abelian normal subgroup on which it acts naturally).
I was recommended to look into the book "Finite Simple Groups - An Introduction to Their Classification" by Daniel Gorenstein, and indeed, it answers my question to its full extent! Janko's approach is actually quite different from what I thought. In particular, there is no construction of a canonical action of $J_1$ on some vector space derived from the internal structure of the group. I will briefly sketch the explanation given in section 2.3 of the book.
The hypothetical group $J_1$ was abstractly defined as a finite simple group with abelian Sylow-2-subgroups, possessing an involution centralizer of the form $C_2 \times \mathrm{PSL}(2,5)$. Janko recognized that the character table of $J_1$ is uniquely determined by that properties. With the character table in mind, he was able to compute the order $|J_1| = 2^3 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19$, and he began to study the modular characters for each prime dividing that order. By modular representation theory, he eventually showed that $J_1$ possesses exactly one absolutely irreducible representation of degree 7 over $\mathrm{GF}(11)$. So there must be a (necessarily faithful) representation $J_1 \to \mathrm{GL}(7,11)$.
Janko also used the character theoretic information to derive structural properties of $J_1$. Among many other consequences, he showed that $J_1$ is generated by two elements $y,z$ with orders $o(y) = 7$ and $o(z) = 5$ satisfying several relations. Without loss of generality, we may represent $y$ by a permutation matrix $Y \in \mathrm{GL}(7,11)$ which is unique up to conjugacy. Once $Y$ is fixed, it turns out that there is only one potential candidate $Z \in \mathrm{GL}(7,11)$ (representing $z$) which satisfies all the necessary relations with respect to $Y$. (These are in fact the famous matrices which can be found on Wikipedia.) Thus, the uniqueness of $J_1$ was established.
The existence of $J_1$ was reduced without much difficulty to showing that these two matrices generate a group of order 175560. This task has been carried out by M. A. Ward.