Dear GAP Forum,

I like to add three remarks about the recent mail of Alexander Konovalov:

I have tried to calculate maximal subgroups of symmetric groups
using MaximalSubgroups in GAP 3.4.4.
This was successfull for all n not greater then 8, but for n=9
I obtain the following message:

Error, sorry, can' t identify the group's solvable residuum in

Let me first mention that this problem stems from a very small list of
perfect groups in GAP3 being available for finding the solvable residuum.
In GAP4 this list has been extended (using the perfect groups as computed by
Plesken and Holt), and so it is possible to compute the lattice of much
larger groups.
Incidentally, in the most recent release 4.2, GAP contains special code for
the perfect subgroups of S_n that should work up to degree 20 or so (though
the total number of subgroups will make it unlikely a lattice computation
will ever finish.)

For degree up to 50, Derek Holt already gave a list of primitive maximal
subgroups for S_n and A_n.

If you use this list, please be aware that the numbering given in his mail
refers to the GAP3 version of the primitive groups library. In GAP4 the
numbering of the primitive groups has changed (please don't complain to me
about this -- this was not my idea. The person who designed the library in
the first place with this feature is now working in industry ...).

I append a list similar to the one given by Derek, but using the GAP4
enumeration. (This list incorporates the corrections mentioned by Derek in
his second forum mail.)

(There is a paper:
@article{liebeckpraegersaxl87,
  author =	 "Martin~W. Liebeck and Cheryl~E. Praeger and Jan
                  Saxl",
  title =	 "A Classification of the Maximal Subgroups of the
                  Finite Alternating and Symmetric Groups",
  journal =	 JALGE,
  volume =	 111,
  year =	 1987,
  pages =	 "365--383"
}
which describes how one gets such lists.)

Finally, permit me the following remark:

Besides being discussed extensively in the manual, the problem also did come
up in earlier forum mails. To find these, one can
use the search feature provided on the St Andrews GAP web page
http://www-gap.dcs.st-and.ac.uk/gap/Search/search.cgi

Searching for ``solvable residuum'' in the forum and the GAP3 manual,
for example, will give you further information about the problem discussed
above.

Best regards,

Alexander Hulpke

Appendix:

indices for maximals of the symmetric group: (GAP4 numbering)
[ [  ], [  ], [  ], [  ], [ 3 ], [ 2 ], [ 4 ], [ 5 ], [ 7 ], [ 7 ], [ 4 ], 
  [ 4 ], [ 6 ], [ 2 ], [  ], [  ], [ 5 ], [ 2 ], [ 6 ], [ 2 ], [ 1, 3, 7 ], 
  [ 2 ], [ 4 ], [ 3 ], [ 22, 26 ], [ 5 ], [ 11 ], [ 12 ], [ 6 ], [ 2 ], 
  [ 8 ], [ 5 ], [  ], [  ], [  ], [ 16, 12 ], [ 9 ], [ 2 ], [  ], [ 4, 6 ], 
  [ 8 ], [ 2 ], [ 8 ], [ 2 ], [ 5 ], [  ], [ 4 ], [ 2 ], [ 33, 38 ], [ 7 ] ]

indices for maximals of the alternating group (duplicate numbers show the
class splits): (GAP4 numbering)
[ [  ], [  ], [ ], [ ], [ 2 ], [ 1 ], [ 5, 5 ], [ 3, 3 ], [ 6, 9, 9 ], 
  [ 6 ], [ 6, 6 ], [ 2, 2 ], [ 5, 7, 7 ], [ 1 ], [ 4, 4 ], [ 11, 11 ], 
  [ 8, 8 ], [ 1 ], [ 5 ], [ 1 ], [ 2, 6 ], [ 1 ], [ 5, 5 ], [ 1, 1 ], 
  [ 21, 24 ], [ 3 ], [ 10, 13, 13 ], [ 6, 6, 11 ], [ 5 ], [ 1 ], 
  [ 7, 9, 9, 10, 10 ], [ 3, 3, 4 ], [ 2, 2 ], [  ], [ 2, 2 ], [ 10, 10, 11 ], 
  [ 8 ], [ 1 ], [  ], [ 3, 5 ], [ 7 ], [ 1 ], [ 7 ], [ 1 ], [ 4, 7, 7 ], 
  [  ], [ 3 ], [ 1 ], [ 32, 37 ], [ 2, 2, 6 ] ]