In terms of the classification of simple groups, it is known that every finite simple group is either:
- Cyclic
- Alternating
- Of Lie type
- One of the 26 Sporadic groups
On the other hand there are 5 Exceptional continuous Lie groups $G_2$, $F_4$, $E_6$, $E_7$, $E_8$. And four infinite families A, B, C and D.
It is also known that some finite sporadic groups are related to Lie groups. (The McKay observation. $E_8$ to the Monster, $E_7$ to the Baby Monster and $E_6$ to the Fischer group.
It is also known that ever pair of division algebras correspond to a Lie group covering most exceptional groups. $(A,B)\rightarrow L$. This is the Freudenthal magic square.
There are more sporadic finite groups than exceptional Lie groups. However, could it be possible that there is a correspondence such that every ordered pair of Lie groups corresponds to a finite group?
$$(A, B)\rightarrow G$$
Interestingly if either A or B or both are from infinite families of Lie groups it would give an infinite family of finite groups. But if both A and B are exceptional this would give 25 sporadic finite groups. It would also correspond to infinite families of all the different Lie types. Which would account for all the different number fields. e.g. $A_n(q^2)$
Most infinite families of finite groups of Lie type correspond to just one Lie group (as far as I know).
If all this happened to be true, it would mean that most sporadic finite groups would be related to a quartet of division algebras $(A, B, C, D)\rightarrow G$. All combinations of unordered quartets of the division algebras $(\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O})$ would give 35 groups.
So the question is, is it possible to create a finite group from a pair of Lie groups? And could they be connected to the sporadic groups?
(Edit: To give some further clues, the Conway groups are related to symmetries of the Leech lattice (24D) which is itself related to triplets of octonions $\Lambda_{24} \subset (\mathbb{O},\mathbb{O},\mathbb{O})$ or three copies of the $E_8$ lattice )
How might this be done? Well for for a pair of division algebras the Lie group is defined by:
$$\mathfrak{der}(A)\oplus\mathfrak{der}(B)\oplus\mathfrak{sa}_3(A\otimes B)$$
So perhaps could you replace the division algebras with Lie algebras and see what happens?
Another more complicated idea to get a group: Take N=4 Super-Yang Mills with gauge group $A$ and compactify on torus given by the lattice from the roots of Lie group $B$. What will the resultant symmetry be?