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Dear GAP Forum,

Alexander Moreto asked

Computing the character table of a large group may, of course, take a very

long time. But if I'm just interested in a few characters (like, for

instance, the irreducible characters of degree n for some fixed n or the

faithful irreducible characters), if there some faster way to compute

them than computing the whole character table?

and Alexander Hulpke answered

There are methods that compute new characters from old, but often it is not

guaranteed that thay will obtain all characters. What you can try is to run

the Dixon-Schneider Algorithm for a few sets (see section `Advanced Methods

for the Dixon-Schneider algorithm' of the reference manual:

http://www.gap-system.org/Manual4/htm/ref/CHAP069.htm#SECT014

)

and then try toi obtain the remaining characters by working with them as

class functions (see chapter ``Class functions'' in the manual), however it

is perceivable that this process will not find all characters you are

interested in, but only a subset. In particular it is hard to guarantee that

one has found all characters of a given degree, or all faithful characters,

unless the whole table has been determined.

It depends on the structure of the group what additional tricks are

available:

If the group is solvable then one can compute the irreducible degrees

with reasonable effort; from that one knows whether one has found

all irreducibles of a given degree.

If the group is abelian-by-supersolvable then one should try to compute

the irreducible characters with the Baum-Clausen method instead of Dixon's

algorithm.

The computations can be accelerated by choosing an appropriate

representation of the group, e.g., a PC group instead of a permutation

group if possible.

If the group has a small nontrivial normal subgroup then one can compute

the characters of the factor group first, lift them to characters

of the whole group, and feed them into the computation of the remaining

characters.

And if related information is stored in the character table library then

one can of course use it.

All the best,

Thomas

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