# where to find the tables of irreducible character of the sporadic simple groups and their automorphism groups?

I need tables of complex irreducible characters of the sporadic simple groups and their automorphism groups $Aut(G)$ for some calculation. Also I need information about maximal subgroups of $Aut(G)$

Where Can I find them ?

I checked ATLAS but I am not able to find anything useful. I guess I am missing something.

For example Take $G=M_{12}$

The page does not have any information about $Aut(G)$, also I am not able to locate character table(complex irreducible )

Edit

I am reading from Inverse Galois Theory by Gunter Malle. Regarding Why I need these tables, see the attached image. He has done all the calculation by using data from ATLAS.

Edits 2

Why these Character Tables are different ?

• A hint for the future: please learn how to copy and paste the text from GAP terminal into your question. If you indent it by four spaces then it will be displayed as a code. This is more productive and useful than screenshots: it's searchable and copyable, what helps other users interested in your question. – Alexander Konovalov Apr 1 '17 at 19:05

Here is some GAP code to compute it, and display it.

gap> M12 := PrimitiveGroup(12,2);
M(12)
gap> A := AutomorphismGroup(M12);
<group with 7 generators>
gap> G := Image(IsomorphismPermGroup(A));
<permutation group of size 190080 with 7 generators>
gap> C := CharacterTable(G);
CharacterTable( <permutation group of size 190080 with 7 generators> )
gap> Display(C);


...

Edit: As Alexander Konovalov has pointed out, you can do this more easily be reading the character table from the library, and for the larger sporadic groups this approach would be much better, because it would take longer to compute them from scratch.

gap> C := CharacterTable("M12.2");
CharacterTable( "M12.2" )

• Any reasons to not to use CharacterTable("M12") to retrieve it from the library and not to compute it on fly? This will also guarantee the same order of characters and conjugacy classes each time it's called. – Alexander Konovalov Mar 28 '17 at 10:47
• Oops, sorry, of course I had CharacterTable("M12.2") in mind. – Alexander Konovalov Mar 28 '17 at 10:55
• @AlexanderKonovalov No there is no reason whatsoever not to do that! – Derek Holt Mar 28 '17 at 10:56
• @DerekHolt What does "M12.2" means. – Tensor_Product Mar 28 '17 at 12:13
• @Tensor_Product: that's exactly what I've said in the comment above. If you're retrieving the table from the library, the ordering is fixed and will be the same each time. If you compute it on fly, it may differ because of some randomised methods being used. – Alexander Konovalov Mar 28 '17 at 14:18

The online version of the ATLAS you linked to is obsolete. You find a more current version at http://brauer.maths.qmul.ac.uk/Atlas/v3/

Regarding your example of $M_{12}$, the page http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/ lists representations, conjugacy classes and maximal subgroups not only for the group $M_{12}$ itself, but also for the automorphism group and their double covers.

• Note, however, that each version of the ATLAS website contains errors, e.g. for v3, look at the order on brauer.maths.qmul.ac.uk/Atlas/v3/exc/E64 -- IMHO a better way to access this is to use the AtlasRep package in GAP, see gap-system.org/Packages/atlasrep.html . It is more reliable (data in is heavily cross-checked) and more up-to-date (contains additional data not on the website). – Max Mar 29 '17 at 7:22

The book "ATLAS of Finite Groups" (which is probably related to the online "ATLAS of Finite Group Representations that you linked to) contains these character tables, including the character table of $M_{12}$:

• What about the Aut(M_{12}). Thanks a lot . – Tensor_Product Mar 28 '17 at 7:22
• @Tensor_Product $M_{12}$ has one nontrivial outer automorphism. The automorphism groups of the sporadic groups are easy to find, for example on wikipedia – user430043 Mar 28 '17 at 7:31
• I read about it, but for calculation purpose I need explicit table of complex irreducible character of $Aut(M_{12})$/ Kindly see the attached image. – Tensor_Product Mar 28 '17 at 7:34
• @Tensor_Product That information is actually in the table above: the characters from $\chi_{16}$ on are actually "generator cohorts of characters" coming from the double cover of $M_{12}.$ The notation is rather complicated but if you can find the book then it is explained in the introduction. I assume that Malle is referring to this book. – user430043 Mar 28 '17 at 7:44
• You could also easily calculate it for your self using GAP or Magma. That's what I would do if I wanted the full character table explicitly – Derek Holt Mar 28 '17 at 8:01