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I am consulting ATLAS of finite group for character table of Automorphism Group of sporadic group.

I am reading from Inverse Galois Theory by G. Malle

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Let me start with $G=M_{12}$

This(image attached) is from ATLAS page of $G=M_{12}$, It's automorphism group is $G.2$

Now in the proposition we have taken conjugacy class triple $(2C,3A,12A)$, but see the block(character table) for $G.2$ on right hand side top, $2C$ and $12A$ are there but there is no $3A$ in that block.

What is the explanation ?

In the character table of $G.2$ there is no conjugacy class of type $3A$ but in proposition he taken the triple $(2C,3A,12A)$.

In all other cases, I am encountering the same kind of problem, eg. In $Aut(M_{22}), Aut(J_{2})$, etc

Kindly sort me out. enter image description here

------------------------Edits ---------------------------- Page 21, The ATLAS

enter image description here

Tensor_Product
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  • I can see a 3A - it's the fourth column of character entries from the left. – Derek Holt Mar 29 '17 at 07:49
  • @DerekHolt But it is not in the character table of $G.2$ – Tensor_Product Mar 29 '17 at 08:58
  • What is G, here? I would have guessed it's M12 itself. – David Roberts Mar 29 '17 at 09:01
  • @DavidRoberts yes, right. Thanks for pointing out, It is confusing. Let me edit the question. – Tensor_Product Mar 29 '17 at 09:02
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    This is just a guess, but perhaps you haven't realised that the character table for $G.2$ includes the character table for $G$. So that conjugacy class 3A for $G$ that Derek mentioned is also a conjugacy class for $G.2$. Would that clarify things? – Nick Gill Mar 29 '17 at 09:11
  • @NickGill But then how one account for the fact that $\chi_{3}$ is not even defined for conjugacy class of type $2C,4C,4D$, etc. (See the empty line.) That table shows(I guess) that out of 15 irreducible character of G, all but $\chi_{3},\chi_{5},\chi_{10},$ are irreducible character of $G.2$ – Tensor_Product Mar 29 '17 at 09:17
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    I suggest you read Section 7 of the ATLAS introduction which tells you how to interpret this sort of (apparently) empty line. Look at, for instance Subsection 15 of that section entitled The detachment of columns for a group $G.2$. – Nick Gill Mar 29 '17 at 09:25
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    $\chi_2+\chi_3$ is a single character of $M_{12}.2$, induced from the character $\chi_2$ (or $\chi_3$) of $M_{12}$. It's value is $0$ on elements not in $M_{12}$. – Derek Holt Mar 29 '17 at 09:33
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    You should also be aware that some pairs of conjugacy classes of $M_{12}$ fuse into a single class of $M_{12}.2$. These pairs are (4A,4B), (8A,8B), and (11A,11B). – Derek Holt Mar 29 '17 at 09:43
  • I got it know, Thanks, Nick, Derek for the help. – Tensor_Product Mar 29 '17 at 11:20
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    It's a pleasure, glad I could help. – Nick Gill Mar 29 '17 at 12:01
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    Nowadays it is maybe a little bit easier to ask GAP to print that character table for you: ct:= CharacterTable( "M12.2" );; Display( ct ); Use AtlasClassNames( ct ); to get the names of the conjugacy classes in the Atlas. So you don't have to derive yourself the character table from the table of $M_{12}$ and information on fusion of classes/ extension of characters, like in the printed Atlas. – Frieder Ladisch Mar 29 '17 at 13:29
  • @FriederLadisch Can I compute the conjugacy class representative of "M12.2" through some command. I know how to find order of class representative and size of conjugacy class but Is there some command by which I can explicitly write the class representative. – Tensor_Product Mar 29 '17 at 13:55
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    Well, if I say ct:= CharacterTable( "M12.2" );, then ct is a table from the Character Table Library of GAP, which has no access to the group ($M_{12}.2$ in this case), and so you can't get class representatives. So you first have to construct "M12.2" as a group, and then tell GAP that ct is the table of that group. See Section 71.6 of the GAP manual. – Frieder Ladisch Mar 29 '17 at 14:15

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