[GAP Forum] Re: a question?

[Thomas Breuer]

Dear Behrooz Khosravi,

you asked

>  I study some properties about sporadic groups and I
> need the size of the normalizers of p-Sylow subgroups
> of sporadic
> 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

More information can be found for example in the following paper.

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

All the best,
Thomas