[GAP Forum] ClassNames and the Atlas

[Thomas Breuer]

Dear GAP Forum,

Simon Nickerson wrote

> Is there any way of getting GAP to use the names of conjugacy classes as
> described in the Atlas of Finite Groups (and the Web Atlas)? The command
> 
> CN := ClassNames(CT, "ATLAS");
> 
> works when CT is the character table of a simple group, but when it is an
> almost-simple group, there is disagreement. For example, the conjugacy
> classes of elements of order 8 in HS.2 are called:
> 
> Atlas:   8A   8BC  8D   8E
> GAP:     8A   8B   8C   8D
> 
> (i.e. GAP labels the classes alphabetically, and the Atlas labels them
> according to how the corresponding classes in the simple group HS fuse
> together). This might cause problems, because there are two conjugacy
> classes of HS.2 which might conceivably be called '8D'.
> 
> If there is no way to get GAP to use Atlas conventions, does anyone have a
> list of the examples where the two systems disagree?

The documentation of `ClassNames' says

    When  `ClassNames'  is  called  with  two  arguments, the second being the
    string  `"ATLAS"',  the  class  names returned obey the convention used in
    Chapter 7, Section 5 of the ATLAS of Finite Groups cite{CCN85}.

This means that this function is *not* thought to take case of ``relative''
class names as in the example above.

For character tables of almost simple groups in the ATLAS,
one can use the function `AtlasClassNames' from the GAP package `AtlasRep'.

  gap> LoadPackage( "AtlasRep" );;
  gap> AtlasClassNames( CharacterTable( "HS.2" ) );
  [ "1A", "2A", "2B", "3A", "4A", "4B", "4C", "5A", "5B", "5C", "6A", "6B", 
    "7A", "8A", "8BC", "10A", "10B", "11AB", "12A", "15A", "20AB", "2C", "2D", 
    "4D", "4E", "4F", "6C", "6D", "6E", "8D", "8E", "10C", "10D", "12B", "14A", 
    "20C", "20D", "20E", "30A" ]

Note that this function is based on a list of the available ATLAS tables,
computing the right names in cases such as $S_6$, $PGL_2(9)$, and $M_{10}$
would be impossible without that.
So it is not defined what the function returns for almost simple non-ATLAS
tables.

All the best,
Thomas