I came across the following paragraph on pp.56 of Marshall Hall’s “The theory of groups” 1959 ed.
... the non-Abelian group of order 6 may be faithfully represented as a transitive permutation group on three letters and also on six letters ...
My question is,
In general, is there a theory about this? That is, given a non-Abelian group of order n, on what conditions it can be faithfully represented by different transitive permutation groups?
If the answer to the above question is yes, how do we call it? Multiple faithful representation? or something else?
Thanks in advance.