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For example in Kluener's data base of transitive subgroups of $S_n$ ( http://www.math.uni-duesseldorf.de/~klueners/minimum/minimum.html ), one can read their name like the one in the title. What informations does it provide, and how to read it ?

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Each of these is a separate name for the same group. Names are not precise, but they can be useful.

The first name says that this group is $2\times S_4$ acting transitively on 6 points. $2\times S_4$ acts intransitively on 6 points as well, but that is a different permutation group that happens to be isomorphic as abstract group.

The second name says this is a semi-direct product of $2^3 = 2 \times 2 \times 2$ acting intransitively on $2+2+2$ points (so a standard direct product) with the symmetric group on 3 points acting naturally by permuting the orbits. In other words, it is also the next name:

The third name says this is a wreath product of the symmetric group on 3 points acting on the cyclic group on 2 points, $2 \wr S_3$.

By the time one looks at transitive groups on 12 points, the names become fairly elaborate in order to distinguish very similar groups, and at 16 points, I believe no one bothers naming them since there are too many with too similar structure.