I think part of the problem that I have been having in classifying the set of operations is that they do not comprise a simple set of objects. Mathematical subjects are partitioned and so we may only get a partial view of something that is by nature more complex. This list of Wikipedia articles may give a more complete picture.
Galois group
Group action (transformation group)
Class (set theory)
Classification theorem
simple groups
algebraic group
Coxeter group
But the articles in a sense seem to be talking around our transformation group for the magic squares. Part of the problem is abstraction or oversimplification or idealization rather than dealing with specific examples.
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