Application of nonfamous finite groups in computer science I have searched a lot about applications of finite groups in computer science. Most of the results include:

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*Finite fields or groups of numbers coprime to $n$ which are widely used in cryptography and coding theory

*Permutations (symmetric group)

*Ring of matrices over an arbitrary field

But group theory is much more enormous and broader than these groups and includes many exotic enormous groups. I wonder if there are some applications of other groups in computer science.
Specifically, I would appreciate if you mention some other finite (or at least finitely generated) groups (not semigroups or monoids) with their applications.
Example) for instance, optimal solving of a Rubik's cube is a computationally-intensive problem (which is called God's algorithm). Another example I want to see is something like monster group or some other finite simple groups.
 A: Automorphism groups of codes. For example, the sporadic Mathieu group $M_{24}$ is the automorphism group of the extended binary Golay code.
A: Isomorphism testing is the area that comes to mind for me. In Graph Isomorphism (GI), the key techniques are Weisfeiler--Leman (WL) and Permutation Group Algorithms. At a high level, Babai uses WL to partition (tuples of) vertices into "small" color classes; after which, he applies permutation group algorithms to decide if there exists a color-preserving isomorphism.
In the multiplication table model, Group Isomorphism (GpI) reduces to GI, but is strictly easier under many-to-one computable reductions that are (for instance) $\textsf{AC}^{0}$-reductions. So while GpI is easier than GI when we zoom in with a microsocope, the best we know how to do in general is still $n^{\Theta(\log n)}$.
There are what I'd consider three main thrusts in GpI for the multiplication table model:

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*Coprime and tame extensions (culminating in this paper https://arxiv.org/abs/1507.01917)

*Groups without Abelian normal subgroups. C.f., Code Equivalence and Group Isomorphism, Polynomial-Time Isomorphism Test for Groups with No Abelian Normal Subgroups, and the follow-up work where the solvable radical $\text{Rad}(G) = Z(G)$, or $Z(G) \lneq \text{Rad}(G)$ but $\text{Rad}(G)$ is elementary Abelian. Algorithms for Group Isomorphism via Group Extensions and Cohomology

*Class $2$ $p$-groups of exponent $p$. See for instance, the works of James Wilson (https://www.math.colostate.edu/~jwilson/products.html).

There are some additional works, such as the Grochow--Qiao Tensor Isomorphism paper (https://arxiv.org/pdf/1907.00309.pdf). Motivation for GpI comes from the relationship between class $2$ $p$-groups of exponent $p$ ($p > 2$). The isomorphism invariant is the commutator subgroup. Testing whether two commutator subgroups are equivalent is precisely testing whether two tensors are pseudo-isomoetric. The Baer correspondence provides that we can reverse the construction: given such a tensor, we can construct a class $2$ $p$-group of exponent $p$ ($p > 2$). The Tensor Isomorphism paper also provides additional applications of algebra in CS.
The papers I've cited do a good job surveying the literature for some results I've omitted, but this should give you a sense of what's out there.
