I was asked how to compute the decomposition of a permutation character for
Fi_{24}' into irreducible ones. I don't really see whether there are
appropriate function(s) to do it. Could somebody give an advise how to do it
in GAP, if it's possible?

It rather depends on the form in which you have the permutation
character. If it's actually as a character with the classes in the
standard order then this should be easy, just recover the character
table with

ct := CharTable("F3+");;

And then you can take inner products:

For example:

List([1..Length(ct.irreducibles)],i->ScalarProduct(ct, permchar,i));

Will return a list giving the multiples of the irreducibles which
contribute to permchar.

If you have explicit permutations for your permutation character
then

a) It must be one of a handfull of known permutation characters
because all the others are too big. You can probably look up the
answer.

b) Otherwise you need to identify the classes of elements in your
permutation group with the standard named classes in F24'. This
problem has been discussed on this forum before. There is no totally
generic simple way, but it's not usually too hard in any particular
case.

Finally, if you just have the subgroup whose cosets you are
permuting (in some sense) your best bet is probably to work out it's
character table (though this is non-trivial for the larger ones) and
then induce the trivial character.

If you are doing a lot of computing in this group, you might want to
get in touch with me directly. I have some special-purpose programs
for computing efficiently in groups of this size, and I have quite a
lot of experience with this particular group.

Steve