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

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  • 1
    $\begingroup$ 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? $\endgroup$ – Nick Gill Mar 29 '17 at 9:11
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    $\begingroup$ 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 $G.2$. $\endgroup$ – Nick Gill Mar 29 '17 at 9:25
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    $\begingroup$ $\chi_2+\chi_3$ 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 $M_{12}$. $\endgroup$ – Derek Holt Mar 29 '17 at 9:33
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    $\begingroup$ 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). $\endgroup$ – Derek Holt Mar 29 '17 at 9:43
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    $\begingroup$ It's a pleasure, glad I could help. $\endgroup$ – Nick Gill Mar 29 '17 at 12:01

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