> < ^ Date: Wed, 14 May 2003 11:23:13 +0100 (BST)
> < ^ From: Derek Holt <dfh@maths.warwick.ac.uk >
< ^ Subject: Re: maximal subgroups

Dear GAP Forum

Thomas Breuer has already posted maximal subgroups of PSL(5,2). I include a
list of representatives of the maximal subgroups of PSL(6,2) (i.e. the
group returned by the GAP command PSL(6,2)) - there are 9 of these.

```I don't have the list to hand for PSL(7,2) =~ GL(7,2), but that is quite easy.
As subgroups of the matrix group GL(7,2), there are 6 irreducible maximals,
two each of the shapes
2^6:L(6,2),  2^10:(L(2,2) X L(5,2))  and  2^12:(L(3,q) X L(4,2))
(where you get the second of each pair as inverse transpose of the first)
```

and also the semilinear group PGammaL(1,2^7).

So, 7 classes altogether.

Derek Holt.

```L62max := [
Group([
(1, 11, 56, 49, 8, 50, 2)(3, 10, 51, 9, 57, 58, 48)(4, 61, 7, 55, 52, 62,
13)(5, 54, 63, 6, 60, 12, 15)(16, 35, 42, 19, 41, 25, 26)(17, 40, 18,
34, 33, 43, 24)(20, 30, 45, 36, 29, 39, 23)(22, 31, 38, 28, 44, 47, 37),
(1, 37, 4, 10, 39, 31, 11, 2, 27)(3, 62, 5, 47, 35, 21, 44, 29, 16)(6, 17,
38, 58, 15, 8, 60, 30, 46)(7, 52, 34, 48, 40, 23, 55, 28, 53)(9, 25, 26,
36, 33, 14, 45, 56, 20)(12, 54, 57, 49, 13, 19, 61, 59, 42)(18, 24, 63,
32, 43, 41, 50, 51, 22)
]),
```
```Group([
(1, 16)(2, 19)(4, 7)(5, 23)(6, 20)(8, 26)(9, 10)(11, 25)(12, 29)(15, 30)(21,
22)(24, 27)(32, 50)(33, 34)(35, 49)(36, 53)(39, 54)(41, 56)(42, 59)(44,
47)(45, 63)(46, 60)(48, 51)(61, 62),
(1, 7, 50, 63)(2, 26)(3, 29, 48, 37)(4, 47, 15, 36)(5, 40, 61, 27)(6, 53,
13, 62)(8, 22, 59, 46)(9, 17)(10, 12, 57, 52)(14, 35, 54, 16)(18, 20,
33, 44)(21, 38, 30, 45)(23, 60, 28, 55)(25, 31, 42, 39)(32, 43)(34,
49)(41, 58)(51, 56)
]),

Group([
(1, 22, 38, 25, 11, 42, 39, 15, 45, 51, 44, 37, 10, 60)(2, 5, 17, 50, 58, 3,
19, 55, 43, 49, 41, 52, 56, 6)(4, 7, 20, 35, 8, 57, 16, 36, 28, 26, 24,
29, 12, 62)(9, 47, 54, 61, 23, 48, 63, 18, 33, 13, 40, 34, 30, 31)(14,
59, 21, 53, 46, 32, 27),
(1, 36, 52, 28, 15, 11, 33, 58, 51, 26, 45, 29, 43, 63, 61, 53, 56, 59, 23,
46, 49, 18, 32, 30, 7, 6, 34, 22, 10, 5, 14, 47, 21, 38, 60, 17, 12, 39,
24, 37, 16, 40, 19, 4, 42, 27, 9, 41, 55, 48, 54, 20, 2, 8, 13, 3, 44,
57, 31, 35, 50, 62, 25)
]),

Group([
(1, 11)(2, 8)(5, 15)(6, 12)(16, 25)(17, 18)(19, 26)(20, 29)(21, 22)(23,
30)(24, 27)(28, 31)(32, 41)(33, 34)(35, 42)(36, 45)(37, 38)(39, 46)(40,
43)(44, 47)(49, 59)(50, 56)(53, 63)(54, 60),
(1, 40, 56, 60, 13, 47, 61, 37, 23)(2, 28, 3, 52, 59, 8, 54, 39, 11)(4, 49,
34, 18, 24, 50, 22, 41, 16)(5, 25, 26, 46, 21, 29, 43, 12, 7)(6, 45, 33,
38, 35, 58, 32, 14, 27)(9, 30, 31, 55, 15, 51, 62, 17, 44)(10, 42, 36,
63, 57, 20, 53, 19, 48)
]),

Group([
(1, 2)(4, 39)(5, 37)(6, 38)(7, 36)(9, 10)(12, 47)(13, 45)(14, 46)(15,
44)(17, 18)(20, 55)(21, 53)(22, 54)(23, 52)(25, 26)(28, 63)(29, 61)(30,
62)(31, 60)(33, 34)(41, 42)(49, 50)(57, 58),
(1, 53, 16, 43, 39, 7, 38, 50, 54, 25, 17, 30, 55, 44)(2, 60, 42, 18, 23,
13, 21, 49, 63, 35, 40, 46, 61, 31)(3, 9, 58, 57, 48, 10, 51)(4, 47, 8,
15, 41, 27, 45, 52, 37, 59, 12, 32, 33, 20)(5, 26, 24, 36, 14, 28, 11,
6, 19, 34, 29, 62, 22, 56)
]),

Group([
(1, 9, 20)(2, 51, 45)(3, 58, 57)(4, 53, 43)(5, 60, 63)(7, 15, 18)(8, 29,
21)(10, 46, 56)(11, 39, 44)(12, 40, 62)(13, 33, 42)(14, 27, 19)(16, 52,
34)(17, 61, 54)(22, 50, 36)(23, 59, 48)(24, 41, 55)(25, 32, 35)(30, 47,
49)(31, 38, 37),
(1, 30, 33, 16, 25, 55)(2, 36, 34, 42, 26, 13)(3, 58)(4, 44, 18, 61, 21,
43)(5, 50, 51, 45, 12, 28)(6, 8, 48, 23, 15, 38)(7, 22, 17)(9, 46, 54,
31, 63, 49)(10, 20, 53, 37, 60, 11)(14, 56, 39, 24, 41, 32)(19, 35, 52,
59, 29, 27)(40, 62, 47)
]),

Group([
(1, 36, 40, 63, 53, 7)(2, 55, 48, 49, 21, 61)(3, 19, 24, 14, 32, 58)(4, 18,
60, 38, 31, 15)(5, 54, 20, 25, 42, 8)(6, 37, 12, 23, 10, 50)(9, 33, 30,
43, 44, 45)(11, 22, 46, 26, 57, 16)(13, 51, 34)(17, 47, 62)(27, 29, 56,
52, 35, 41)(39, 59),
(1, 59)(2, 43)(3, 16)(4, 45)(5, 22)(7, 61)(9, 51)(10, 35)(11, 24)(12,
37)(13, 30)(15, 53)(17, 56)(18, 40)(20, 46)(23, 62)(25, 48)(26, 32)(28,
38)(31, 54)(34, 49)(36, 55)(42, 57)(44, 63)
]),

Group([
(1, 2)(5, 6)(8, 11)(12, 15)(17, 18)(21, 22)(24, 27)(28, 31)(33, 34)(37,
38)(40, 43)(44, 47)(49, 50)(53, 54)(56, 59)(60, 63),
(1, 36, 48, 60, 22, 7)(2, 11, 43, 53, 54, 25)(3, 47, 27, 9, 32, 30)(4, 46,
63, 57, 28, 8)(5, 10, 15)(6, 37, 20, 12, 42, 17)(13, 14, 33, 58, 51,
19)(16, 34, 21, 40, 26, 45)(18, 41, 62, 29, 44, 52)(23, 35, 49, 24, 38,
59)(31, 39)(50, 55, 61)
]),
```
```  Group([
(4, 37)(5, 36)(6, 39)(7, 38)(8, 41)(9, 40)(10, 43)(11, 42)(20, 53)(21,
52)(22, 55)(23, 54)(24, 57)(25, 56)(26, 59)(27, 58),
(1, 55, 48, 2, 52, 51)(4, 49, 53)(5, 6)(7, 50, 54)(8, 36, 29, 12, 21, 40)(9,
19, 45, 14, 33, 27)(10, 16, 46, 13, 34, 24)(11, 39, 30, 15, 22, 43)(17,
25, 61, 32, 44, 57)(18, 26, 62, 35, 47, 58)(20, 31, 56, 38, 41, 63)(23,
28, 59, 37, 42, 60)
])
];
```

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