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.)

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    $\begingroup$ I just want to note that even if you know all of the normal subgroups of $G$ and all of $G$'s conjugacy classes, you still might not have enough information to construct the character table. $G$ could be simple, for instance. $\endgroup$ – John McHugh Mar 27 '18 at 18:02

Since you know that $G/[G,G]$ has order $2$, the two $1$-dimensional characters are just the trivial one and the sign representation.

$S_4$ has two $2$-transitive permutation representations, its natural one om four points, and also one on three points coming from its quotient group isomorphic to $S_3$. You get $3$- and $2$-dimensional irreducible representations from these, and the associated characters can be easily written down as $1$ less than the number of fixed points of the elements.

You can multiply both of those by the sign representation, which gives another $3$-dimensional irreducible. (The $2$-dimensional one stays the same.)

So now you have them all.


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