I am interested in this paper which I can't read because it's in German:

*Frobenius, G.*, Über die Charaktere der mehrfach transitiven Gruppen., Berl. Ber. 1904, 558-571 (1904). ZBL35.0154.02.

A free online copy is here. I am specifically interested in the character table of $M_{12}$ which appears at the top of p.568.

I would like to know what, exactly, Frobenius proved with regard to this character table. If I understand correctly, at the time of publication, the existence of $M_{12}$ was still somewhat contentious. Mathieu had written down some permutations that generated a 5-transitive group on 12 letters, but there was some uncertainty as to whether or not these permutations generated all of $A_{12}$. I believe the uncertainty continued until the 1930's, at which point Witt used Steiner systems, to show that $M_{12}$ was really a **proper** subgroup of $A_{12}$.

In light of this, my guess is that Frobenius proved the following theorem:

Theorem. Suppose that $G$ is a sharply 5-transitive subgroup of $S_{12}$. Then the character table of $G$ is as follows...

Note that the theorem as stated **does not** refer to the specific construction of $M_{12}$ given by Mathieu, but is more a theorem of the kind "Should such a group exist, then...." (a type of theorem that appeared many times in the following 100 years as part of the classification).

However, perhaps I am wrong. So my questions are, specifically:

- Does Frobenius' character table calculation pertain to any sharply 5-transitive subgroup of $S_{12}$, or is it specific to $M_{12}$?
- Does Frobenius give a proof of his calculations, or is it more of a sketch?
- The same questions as above also apply to $M_{24}$, with obvious edits.

**Edit**: Thank you for the useful comments and answers. Perhaps I should add a fourth question to clarify exactly what I am looking for:

- Does Frobenius use any properties of $M_{12}$, apart from sharp 5-transitivity, in his calculation of the character table?

**Edit 2 -- 26 June 2018:** My MMath student, Sam Hughes, and I have just uploaded a preprint connected to this question. In it we write down the character table of a sharply 5-transitive subgroup of $A_{12}$ without making any reference to the Mathieu groups. So, even though this is **not** what Frobenius did (as Frieder Ladisch's answer below makes clear), it is interesting to know that he could have done it if he had wanted to! We have cited Frieder's answer below in the preprint, as it was very helpful for clarifying the history of this work. Thanks again!

orderof these groups was not "properly known", it's hard to imagine what other properties Frobenius might have made use of... $\endgroup$ – Nick Gill Feb 27 '18 at 10:19