# [GAP Forum] Re: a question?

Thomas Breuer thomas.breuer at math.rwth-aachen.de
Thu Feb 26 18:27:53 GMT 2004

```Dear Behrooz Khosravi,

> need the size of the normalizers of p-Sylow subgroups
> groups. Only for Mathieu groups I could use
> g:=MathieuGroup(n);
> and then I compute the size of the normalizers of
> p-Sylow
> subgroups of it. I would be very thankful if you
> kindly let me
> know how I can compute these numbers for 26 sporadic
> simple group.

Many character tables of Sylow normalizers in the sporadic simple
groups are contained in the GAP Character Table Library.
So you can get part of this information with a little loop,
as follows.

for name in AllCharacterTableNames( IsSporadicSimple, true ) do
t:= CharacterTable( name );
primepowers:= Collected( Factors( Size( t ) ) );
Print( name, ": ", primepowers, "\n" );
for pair in primepowers do
p:= pair[1];
if pair[2] = 1 then
# If the prime divides just once then the order of the Sylow
# normalizer can be derived from the table.
orders:= OrdersClassRepresentatives( t );
pos:= Position( orders, p );
size:= SizesCentralizers( t )[ pos ] * ( p - 1 ) /
Number( orders, x -> x = p );
Print( p, ": ", size, "\n" );
else
# We use the table of the Sylow normalizer if it is available.
s:= CharacterTable( Concatenation( name, "N", String( p ) ) );
if s <> fail then
Print( p, ": ", Size( s ), "\n" );
fi;
fi;
od;
Print( "\n" );
od;

This yields something similar to the following output.

B: [ [ 2, 41 ], [ 3, 13 ], [ 5, 6 ], [ 7, 2 ], [ 11, 1 ], [ 13, 1 ],
[ 17, 1 ], [ 19, 1 ], [ 23, 1 ], [ 31, 1 ], [ 47, 1 ] ]
7: 28224
11: 13200
13: 3744
17: 1088
19: 684
23: 506
31: 465
47: 1081

Co1: [ [ 2, 21 ], [ 3, 9 ], [ 5, 4 ], [ 7, 2 ], [ 11, 1 ], [ 13, 1 ],
[ 23, 1 ] ]
3: 157464
5: 10000
7: 3528
11: 660
13: 1872
23: 253

Co2: [ [ 2, 18 ], [ 3, 6 ], [ 5, 3 ], [ 7, 1 ], [ 11, 1 ], [ 23, 1 ] ]
2: 262144
3: 23328
5: 12000
7: 336
11: 110
23: 253

Co3: [ [ 2, 10 ], [ 3, 7 ], [ 5, 3 ], [ 7, 1 ], [ 11, 1 ], [ 23, 1 ] ]
2: 1024
3: 69984
5: 6000
7: 252
11: 110
23: 253

F3+: [ [ 2, 21 ], [ 3, 16 ], [ 5, 2 ], [ 7, 3 ], [ 11, 1 ], [ 13, 1 ],
[ 17, 1 ], [ 23, 1 ], [ 29, 1 ] ]
5: 28800
7: 12348
11: 1320
13: 2808
17: 272
23: 253
29: 406

Fi22: [ [ 2, 17 ], [ 3, 9 ], [ 5, 2 ], [ 7, 1 ], [ 11, 1 ], [ 13, 1 ] ]
3: 78732
5: 2400
7: 252
11: 110
13: 78

Fi23: [ [ 2, 18 ], [ 3, 13 ], [ 5, 2 ], [ 7, 1 ], [ 11, 1 ], [ 13, 1 ],
[ 17, 1 ], [ 23, 1 ] ]
7: 5040
11: 440
13: 468
17: 272
23: 253

HN: [ [ 2, 14 ], [ 3, 6 ], [ 5, 6 ], [ 7, 1 ], [ 11, 1 ], [ 19, 1 ] ]
7: 2520
11: 220
19: 171

HS: [ [ 2, 9 ], [ 3, 2 ], [ 5, 3 ], [ 7, 1 ], [ 11, 1 ] ]
2: 512
3: 288
5: 2000
7: 42
11: 55

He: [ [ 2, 10 ], [ 3, 3 ], [ 5, 2 ], [ 7, 3 ], [ 17, 1 ] ]
2: 1024
3: 216
5: 1200
7: 6174
17: 136

J1: [ [ 2, 3 ], [ 3, 1 ], [ 5, 1 ], [ 7, 1 ], [ 11, 1 ], [ 19, 1 ] ]
2: 168
3: 60
5: 60
7: 42
11: 110
19: 114

J2: [ [ 2, 7 ], [ 3, 3 ], [ 5, 2 ], [ 7, 1 ] ]
2: 384
3: 216
5: 300
7: 42

J3: [ [ 2, 7 ], [ 3, 5 ], [ 5, 1 ], [ 17, 1 ], [ 19, 1 ] ]
2: 384
3: 1944
5: 60
17: 136
19: 171

J4: [ [ 2, 21 ], [ 3, 3 ], [ 5, 1 ], [ 7, 1 ], [ 11, 3 ], [ 23, 1 ],
[ 29, 1 ], [ 31, 1 ], [ 37, 1 ], [ 43, 1 ] ]
3: 864
5: 26880
7: 2520
11: 319440
23: 506
29: 812
31: 310
37: 444
43: 602

Ly: [ [ 2, 8 ], [ 3, 7 ], [ 5, 6 ], [ 7, 1 ], [ 11, 1 ], [ 31, 1 ],
[ 37, 1 ], [ 67, 1 ] ]
2: 256
3: 69984
5: 250000
7: 1008
11: 330
31: 186
37: 666
67: 1474

M: [ [ 2, 46 ], [ 3, 20 ], [ 5, 9 ], [ 7, 6 ], [ 11, 2 ], [ 13, 3 ],
[ 17, 1 ], [ 19, 1 ], [ 23, 1 ], [ 29, 1 ], [ 31, 1 ], [ 41, 1 ],
[ 47, 1 ], [ 59, 1 ], [ 71, 1 ] ]
11: 72600
13: 632736
17: 45696
19: 20520
23: 6072
29: 2436
31: 2790
41: 1640
47: 2162
59: 1711
71: 2485

M11: [ [ 2, 4 ], [ 3, 2 ], [ 5, 1 ], [ 11, 1 ] ]
2: 16
3: 144
5: 20
11: 55

M12: [ [ 2, 6 ], [ 3, 3 ], [ 5, 1 ], [ 11, 1 ] ]
2: 64
3: 108
5: 40
11: 55

M22: [ [ 2, 7 ], [ 3, 2 ], [ 5, 1 ], [ 7, 1 ], [ 11, 1 ] ]
2: 128
3: 72
5: 20
7: 21
11: 55

M23: [ [ 2, 7 ], [ 3, 2 ], [ 5, 1 ], [ 7, 1 ], [ 11, 1 ], [ 23, 1 ] ]
2: 128
3: 144
5: 60
7: 42
11: 55
23: 253

M24: [ [ 2, 10 ], [ 3, 3 ], [ 5, 1 ], [ 7, 1 ], [ 11, 1 ], [ 23, 1 ] ]
2: 1024
3: 216
5: 240
7: 126
11: 110
23: 253

McL: [ [ 2, 7 ], [ 3, 6 ], [ 5, 3 ], [ 7, 1 ], [ 11, 1 ] ]
2: 128
3: 5832
5: 3000
7: 42
11: 55

ON: [ [ 2, 9 ], [ 3, 4 ], [ 5, 1 ], [ 7, 3 ], [ 11, 1 ], [ 19, 1 ], [ 31, 1 ]
]
2: 512
3: 25920
5: 720
7: 8232
11: 110
19: 114
31: 465

Ru: [ [ 2, 14 ], [ 3, 3 ], [ 5, 3 ], [ 7, 1 ], [ 13, 1 ], [ 29, 1 ] ]
2: 16384
3: 432
5: 4000
7: 168
13: 624
29: 406

Suz: [ [ 2, 13 ], [ 3, 7 ], [ 5, 2 ], [ 7, 1 ], [ 11, 1 ], [ 13, 1 ] ]
2: 24576
3: 34992
5: 600
7: 504
11: 110
13: 78

Th: [ [ 2, 15 ], [ 3, 10 ], [ 5, 3 ], [ 7, 2 ], [ 13, 1 ], [ 19, 1 ],
[ 31, 1 ] ]
2: 32768
5: 12000
7: 7056
13: 468
19: 342
31: 465

R. A. Wilson,
The McKay conjecture is true for the sporadic simple groups,
J. Algebra 207 (1998), 294-305.

All the best,
Thomas

```