I study applications of group actions and got a lot of information about nontrivial normal subgroups of a group.
Now I want to construct the character table of $S_4$. Is there a way to get the table of a group (only) with the knowledge of all of its normal subgroups/ number of conjugation classes? (I want to get the table with results which I got when letting groups acting on sets-e.g.: The number of conjugation classes is the number of the conjugation-operation-orbits.)