> < ^ Date: Tue, 28 Mar 2000 14:18:50 +0100 (BST)
> < ^ From: Derek Holt <dfh@maths.warwick.ac.uk >
> < ^ Subject: Re: Maximal subgroups of SymmetricGroup(9)

Dear GAP Forum,

Alexander B. Konovalov wiote:

Dear Forum,
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

etc.

Here is a method for getting the maximal subgroups of the symmetric groups
of degree n up to n=50.

These are of three types:

```(i) Intransitive subgroups, S(r) X S(n-r) for 1 <= r < n/2.

(ii) Imprimitive subgroups, S(r) Wr S(n/r) for all divisors r of
n with 1 < r < n.
```

(iii) Primitive maximal subgroups. These can be found using the command
M := PrimitiveGroup(n,r);
but you have to know the correct values of r.

If I have not made any mistakes, then the list maxprim below gives you this
information.
For 1 <= n <= 50, maxprim[n][1] is the list of those r for which
PrimitiveGroup(n,r) is maximal in SymmetricGroup(n), but NOTE THAT THIS DOES
NOT INCLUDE AlternatingGroup(n), so you have to add that in separately.

(maxprim[n][2] is the list of r for which PrimitiveGroup(n,r) is maximal in
AlternatingGroup(n) and maxprim[n][3] is the subset of maxprim[n][2]
consisting of those that split into two conjugacy classes in the alternating
group. This enables you to find the maximals of AlternatingGroup(n) as well.
Just intersect the groups in (i) and (ii) with the alternating groups,
but note the one exception: (S(2) Wr S(4)) intersected with A(8) is not
maximal in A(8) - it is contained in AGL(3,2). If anyone finds any mistakes
in anything I have written here, I would be very grateful if they could let
me know!)

``` maxprim := [
[ [], [], [] ], 						# 1
[ [], [], [] ], 						# 2
[ [], [], [] ], 						# 3
[ [], [], [] ], 						# 4
[ [3], [2], [] ], 						# 5
[ [2], [1], [] ], 						# 6
[ [4], [5], [5] ], 						# 7
[ [4], [5], [5] ], 						# 8
[ [7], [6,9], [9] ], 					# 9
[ [7], [6], [] ],	 					# 10
[ [4], [6], [6] ],	 					# 11
[ [2], [4], [4] ],	 					# 12
[ [6], [5,7], [7] ], 					# 13
[ [2], [1], [] ],	 					# 14
[ [], [4], [4] ],	 					# 15
[ [], [20], [20] ],	 					# 16
[ [5], [4,8], [8] ], 					# 17
[ [2], [1], [] ],	 					# 18
[ [6], [5], [1] ],	 					# 19
[ [2], [1], [] ],	 					# 20
[ [1,3,7], [2,6], [] ], 					# 21
[ [2], [1], [] ],	 					# 22
[ [4], [5], [5] ],	 					# 23
[ [2], [3], [3] ],	 					# 24
[ [23,26], [21,24], [] ],					# 25
[ [5], [2], [] ],						# 26
[ [13], [11,12], [11] ],					# 27
[ [11], [9,12], [12] ],					# 28
[ [6], [5], [] ],						# 29
[ [2], [1], [] ],	 					# 30
[ [8], [9,10], [9,10] ],					# 31
[ [4], [3,5], [5] ],					# 32
[ [], [2], [2] ],	 					# 33
[ [], [], [] ],	 					# 34
[ [], [4], [4] ],	 					# 35
[ [16,19], [14,20], [20] ],					# 36
[ [9], [8], [] ],	 					# 37
[ [2], [1], [] ],	 					# 38
[ [], [], [] ],	 					# 39
[ [4,6], [2,5], [] ],					# 40
[ [8], [7], [] ],	 					# 41
[ [2], [1], [] ],	 					# 42
[ [8], [7], [] ],	 					# 43
[ [2], [1], [] ],	 					# 44
[ [5], [4,7], [7] ],					# 45
[ [], [], [] ],	 					# 46
[ [4], [3], [] ],	 					# 47
[ [2], [1], [] ],	 					# 48
[ [34,38], [32,36], [] ],					# 49
[ [6], [2,7], [7] ]						# 50
];
```

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