Dear GAP Forum,

Dima Pasechnik wrote

Looking at the field "text" of
CharTable("3^(1+10):U5(2):2"),
on sees the following:
"table computed with CliffordTable( U5(2).2 ->
  3^10:U5(2).2  -> 3^1+10:U5(2).2 )"

So it seems that this table was computed using a
(semi) automated procedure.
In fact, GAP has the function CliffordTable, unfortunately
undocumented, AFAIK.

Could perhaps someone who knows about this functions
be so kind to explain how it can be used (or at least tell,
who is to ask) ? As a matter of
fact, we need to compute the character table of
3^1+10:(2xU5(2).2), so we are wondering if
CliffordTable is the right thing to look at in this
respect.

Indeed the character table of 3^(1+10):U5(2):2 has been
computed using Clifford matrices
(also called Fischer matrices by some authors).
The ideas of this approach can be found in

@proceedings{MR91,
title        = "Representation theory of finite groups and
                finite--dimensional algebras",
booktitle    = "Representation theory of finite groups and
                finite--dimensional algebras",
editor       = "G.~O. Michler and
                C.~R. Ringel",
series       = "Progress in Mathematics",
volume       = "95",
year         = "1991",
publisher    = Birkhaeus,
}

Although there are GAP functions that help to construct
character tables along these lines (We would not call the
procedure semi-automated, at most semi-semi-automated),
this code is not documented because it has not been tested
thoroughly enough.

The Clifford matrices method would in principle be the right
method to compute a table such as that of 3^1+10:(2xU5(2).2),
but we cannot recommend the GAP code mentioned above for use.

So if one really wants to compute this character table,
it is perhaps worth a try to use the Dixon-Schneider method,
since computers are getting faster and larger.
More precisely, one could follow use the functions described
in the manual section ``Advanced Methods for Dixon-Schneider
Calculations'' (and the example in the following section).

Another question is whether one needs to compute the
character table at all.
Probably the group in question is the 3B normalizer in the
Fischer group Fi24, and if one would be interested for example
only in the permutation character of Fi24 corresponding to the
action on the cosets of the 3B normalizer
then the known table of the intersection with the simple group
Fi24' would suffice.

Thomas Breuer and Alexander Hulpke