This may be a very simple question, but I do not know how to approach it.
Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group acting on G (in other words, isomorphic to a permutation group, a permutation group just being a subgroup of a symmetric group).
What I am curious about is:
Is every permutation group isomorphic to some 'familiar' group?
(Feel free to read as "finite permutation group" if it makes a difference.)
What I mean by a 'familiar' group is a group that isn't typically viewed as a group of permutations, such as cyclic groups, dihedral groups, groups of matrices like the general linear group, etc. Phrased another way, is there a 'nice' description of every permutation group in terms of common groups and not depending on a presentation with generators and relations or reference to the object that the permutation groups acts on?
It seems there should be some hope for this, since a fundamental notion of groups is that they describe symmetries. But perhaps this does not mean that every permutation group is a 'nice enough' or 'interesting enough' permutation that it receives a nice name to go by.
I really hope my question is clear, even if the language may not be the best with which to describe what I'm after.
What I have so far is that if our permutation group is abelian, it seems it should be easy to classify it by the fundamental theorem of finitely generated abelian groups, thus providing an answer in that case.
In addition, some subgroups of symmetric groups are recognized as familiar groups, as shown here.
Thanks in advance for the input.
EDIT: What I want to know, if this is any better (and maybe I should have stated it this way in the first place), is just if there are any permutation groups for which the sole, or by far best, description is as a group of permutations, instead of in terms of any kind of combination of groups that do have a description not in terms of the set of permutations on some object.