> < ^ Date: Mon, 05 Aug 1996 14:11:00 +0100 (MET)
> < ^ From: Thomas Breuer <Thomas.Breuer@Math.RWTH-Aachen.DE >
^ Subject: Re: Class Multiplication Coefficients

Dear GAP-Forum,

Anatolii V. Rukolaine asked a question about class multiplication

it's necessary for me to have "Class Multiplication Coefficients" in
an algebra of classes of some finite permutation group G.
Of course, I can use GAP-command "ClassMultCoeffCharTable", but in this
GAP-command it's necessary to point out some "CharTable".
How can I avoid an use of "CharTable"?
Yesterday I wrote a GAP-function for a calculation of Class Multiplication
Coefficients for the group Mathieu M12. I use a straightforward algorithm.
Let a1 and a2 be elements from group G. Let CCl be the conjugacy class with
element a2. With help of GAP-command Elements(CCl) I look over all
elements a3 from conjugacy class CCl and calculate a product a1*a3.
For all a1*a3 I find a conjugacy class with a1*a3 with help GAP-command
"IsConjugate". After the calculation of multiplicity for all the conjugacy
classes I can receive Class Multiplication Coefficients by means of simple
arithmetic with orders of the conjugacy classes.
My IBM-computer has a number GAPstones equal 3223.
For two conjugacy classes with 396 and 495 elements from M12 my computer
calculated the Class Multiplication Coefficients about a half of an hour.
For two conjugacy classes with 9504 and 11880 elements from M12 my computer
calculated the Class Multiplication Coefficients more than eight hours
(the whole night) and nothing.
At morning I stoped the calculations. Alas! And so my second question:
how can I calculate Class Multiplication Coefficients more fast?

I do not understand why one wants to avoid dealing with character tables.
The table of M12 is contained in the character table library of GAP,
and there is no problem to identify the columns of that table with the
classes of the group.

If one does not want to use the character table library,
one can compute the character table of M12 easily;
given M12 as a permutation group on 12 points, a machine with about
80000 GAPstones needs less than half a minute for the whole computation,
including the calculation of conjugacy classes (about 7 seconds)
and finally all class multiplication coefficients (about 3 seconds).

In fact GAP uses some class multiplication coefficients to compute the
character table. A description how to access the relevant routines
to compute class multiplication coefficients will be given by
Alexander Hulpke.

My question is:
Are there really good reasons to avoid computing the character table?

Kind regards
Thomas Breuer

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