Dear Mrs. and Mr. Forum,
Thierry Dana-Picard asks the following question.
Let <t> be the character table of a group <g>, given by GAP. Is there a
way to get the elements of the various conjugacy classes of <g> and the
correspondence between these classes and the columns of <t>?
If <t> has been computed from <g> using 'CharTable' then the conjugacy
classes of <g> are stored in the component '<g>.conjugacyClasses', and
their ordering is the same as that of the columns in <t>.
If <t> is a library table then there is no particular representation of
the underlying group <g> attached to <t>, so it is not clear what it means
to ``get the elements'' of a conjugacy class corresponding to a particular
column of <t>. So the problem might be to identify the columns of a
library table <t> with the conjugacy classes of a group <g> for which <t>
is the character table. Probably many of the classes of <g> will be easily
identified by their element and centralizer orders, and maybe other
invariants. If this is not sufficient, one can compute the character
table of <g>, and then find permutations of columns and rows that transform
this table into <t> (e.g. using 'TransformingPermutationsCharTables').
In some cases there is a generic description of conjugacy classes, e.g.,
the classes of symmetric groups are parametrized by partitions, and the
classes of general linear groups are parametrized by normal forms of
matrices. This may make it easier to identify the columns and the classes.