> < ^ Date: Tue, 08 Feb 2000 11:29:36 +0100 (CET)
> < ^ From: Herbert Pahlings <herbert.pahlings@math.rwth-aachen.de >
< ^ Subject: Re: Groups with identical character tables

gap forum gap-forum.

> Does anyone out there have some kind of library of
> groups with identical character tables? I have
> constructed the groups from Mattarei's thesis, but
> I could use some more examples.
Dear Scott Murray,

I am not quite sure what is meant by ``identical character tables'' in this
context, probably just that bijections can be put up so that corresponding
character values coincide. I don't have a library of groups of this sort,
alhough using ``TransformingPermutations'' in GAP one could construct such a
library. The problem is that it certainly will be huge, as you already can
see form the following list for 2-groups of order up to 2^8

order |  number of isomorphism classes | number of character tables \\    
16    |              14                |            11
32    |              51                |            35
64    |             267                |           146
128   |            2328                |           904
256   |           56092                |          8444

In fact among these groups there are not just pairs, but also triples,
quadruples, even larger families of groups having the same character tables.
There is
even a family of 1528 pairwise non-isomorphic groups of order 256 (with rank 5
and elementary abelian commutator factor group and center of order 8)
having the same character table. \\

If you require that the bijections between conjugacy classes 
respect element orders and more
generally the power-maps, i.e. if you are looking for ``Brauer pairs''
you get a far smaller list. In fact, the first examples of such pairs 
were found by Dade (J. Algebra {\bf 1} (1964), 1--4), they are
$p$-groups of exponent $p$ and order $p^7$  defined for $p \geq 5$.
The smallest examples of Brauer pairs were found by E. Skrzipczyk, a former
student of mine, these are the 2-grooups of order 2^8 which have the numbers
( G_{1734}, G_{1735} ), ( G_{1736}, G_{1737} ),( G_{1739}, G_{1740} ),
( G_{1741}, G_{1742} ),( G_{3378},G_{3380} ),$ \\
$ (G_{3379}, G_{3381} ),(G_{3678}, G_{3679}),( G_{4154}, G_{4157} ),
( G_{4155}, G_{4158} ) , ( G_{4156}, G_{4159} ),
in the GAP-library (see also K.Lux,H.Pahlings,
Computational aspects of representation
theory of finite groups II, in: Algorithmic Algebra and Number Theory,
ed. B.H.Matzat, G.-M.Greuel,G.~Hiss, 381--397, 1999).

With best wishes

Herbert Pahlings

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