David Sibley asked in his message from Apr 22 1992:

Is there some easy way to identify from the character table record
which columns are inverses of which other columns? I'd like to be able
to do this to, say, compute class multiplication constants
automatically (by a pre-defined function) from the table. Everything
else needed seems to be readily available in the record.

There is a GAP function 'InverseClassesCharTable' that returns a list
where at position <i> the position of the inverse class of class <i>
is stored. The power maps of the table are in general not sufficient
to compute the inverse classes, so this is done using the 'irreducibles'
component of the table record. We do not store this information on the
table, and do not use it (although it could be used for example by the
scalar product) in order to avoid inconsistencies between character values
and a list of inverses computed at an earlier stage.

Class multiplication constants, or matrices of them, can be computed using
'ClassMultCoeffCharTable' and 'MatClassMultCoeffsCharTable'.

Thomas Breuer