Many commutative groups could be imagined as something "concrete", for example $\mathbb N$ as an abstraction of operations on sets of objects, and from this $\mathbb Z, \mathbb Q$ and $\mathbb R$ are constructed; also if I have some objects and rules to combine them, something like an algebraic structure arises. This is one thing to see it.
Another is more abstract, and views algebraic objects as permitted actions on other objects, in the group setting I mean permutation groups (as this view corresponds to mappings, I think the associativity is crucial here; and given that, in essence both views are equivalent in that a mapping, or set thereof, could be regarded as an object of its own).
Now when I think about non-commutative groups, like $\mathcal S_3$ and many others, just their realisation as permutation groups, i.e. symmetrie groups of other objects, or by explicitly listing them by generators and relations came to my mind.
But are their any concrete, real world examples where non-commuting finite groups are realised as objects on their own (i.e. without seeing them as operations on another object)? What I mean is where in some sense the objects are "experienceable" in the sense that they exist materially, and there are certain operations to combine them?
I hope my question is not to vague and it is clear what I am asking about?!
I think that finite symmetry groups do qualify, e.g., the dihedral groups $D_n$. Polygons are "real", although they might not "exist materially". However, groups are rarely concrete without "another object". I think of symmetry groups of codes, for example. The Mathieu groups arising by the Golay codes. This is great ! Why should I expect that the Mathieu groups appear more concrete in nature ? Rubin's cube does exist materially, and hence a lot of finite non-abelian groups with it - see this question.
References: For a similar question, see this MO question.