TITLE: Multiple faithful representation?
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: Geoff Robinson's answer is perfect, of course, but I submit this answer to address your comment below Geoff's answer.  (I can't write this as a comment because I want to include a picture.)
As Geoff points out, there are many non-trivial examples other than the example Hall presents.  Consider the smallest non-abelian simple group, $A_5$.  It is simple -- i.e. has no non-trivial normal subgroups -- so, for the reasons given by Geoff, every (non-trivial) permutation representation is faithful.
Here is a table (excerpted from my GapNotes.pdf) of all the permutation representations of $A_5$:  

By the way, my profile picture happens to be the subgroup lattice of $A_5$.  So, by looking at it, you can see the congruence lattice -- or "systems of imprimitivity," as the group theorists say -- of each of these representations; it's the interval above the given subgroup.  The picture also makes it apparent that there are three primitive representations (since there are three conjugacy classes of maximal subgroups).