Maths - Sparodic Groups

These are very large groups, for example the smallest Mathieu11 has order (size) of 7920 so a Cayley table for this group would have 7920 rows and 7920 columns. Therefore I will not show these tables in full.

Sparodic Groups

Designation Other terms used Name Order (number of elements)
M11   Mathieu11 7920 = 24*32*5*11
M12   Mathieu12 95040 = 26*33*5*11
M22   Mathieu22 443520 = 27*32*5*7*11
M23   Mathieu23 10200960 = 27*32*5*7*11*23
M24   Mathieu24 244823040 = 210*33*5*7*11*23
J1 HJ or HJM   175560 = 23*3*5*7*11*19
J2   Janko2 604800 = 27*33*52*7
J3     27*35*5*17*19
J4     221*113*33*23*29*31*37*43
HS   Higman-Sims 29*32*53*7*11
McL   McLaughlin 27*36*53*7*11
He F7 Held 210*33*52*7*17
Ru   Rudvalis 214*33*53*7*13*29
Suz   Suzuki 213*37*52*7*11*13
O'N   O'Nan 29*34*5*73*11*19*31
Ly   Lyons 28*37*56*7*11*31*37*67
Co1   Conway  
Co2      
Co3      
Fi22   Fischer  
Fi23      
Fi24 Fi24′    
HN F5 Harada-Norton  
Th F3 Thompson  
B F2 Baby Monster  
M F1 Fischer-Griess Monster  

Generating a Sparodic Groups using a Program

We can use a computer program to generate these groups, here I have used Axiom/FriCAS which is described here.

m11 := mathieu11()
                  
                  
   (1)  <(1 10)(2 8)(3 11)(5 7),(1 4 7 6)(2 11 10 9)>
                                          Type: PermutationGroup(Integer)
order(m11)        
                  
                  
   (2)  7920      
                                                    Type: PositiveInteger
m12 := mathieu12()
                  
                  
   (3)  <(1 2 3 4 5 6 7 8 9 10 11),(11 12)(1 6 5 8 3 7 4 2 9 10)>
                                          Type: PermutationGroup(Integer)
order(m12)        
                  

   (4)  95040
                                                    Type: PositiveInteger
m22 := mathieu22()

   (5)
   <  
       (1 2 4 8 16 9 18 13 3 6 12)(5 10 20 17 11 22 21 19 15 7 14)
    ,
       (3 15)(10 16)(1 2 6 18)(5 8 21 13)(7 9 20 12)(11 19 14 22)
     >
                                          Type: PermutationGroup(Integer)
order(m22)


   (6)  443520
                                                    Type: PositiveInteger
m23 := mathieu23()


   (7)
   <
       (1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23)
    ,
       (2 16 9 6 8)(3 12 13 18 4)(7 17 10 11 22)(14 19 21 20 15)
     >
                                          Type: PermutationGroup(Integer)
order(m23)


   (8)  10200960
                                                    Type: PositiveInteger
m24 := mathieu24()


   (9)
   <
       (1 16 10 22 24)(2 12 18 21 7)(4 5 8 6 17)(9 11 13 19 15)
    ,
       (2 10)(23 24)(1 22 13 14 6 20 3 21 8 11)(4 15 18 17 16 5 9 19 12 7)
     >
                                          Type: PermutationGroup(Integer)
order(m24)


   (10)  244823040
                                                    Type: PositiveInteger
j2 := janko2()


   (11)
   <
       (2 3 4 5 6 7 8)(9 10 11 12 13 14 15)(16 17 18 19 20 21 22)
         (23 24 25 26 27 28 29)(30 31 32 33 34 35 36)(37 38 39 40 41 42 43)
         (44 45 46 47 48 49 50)(51 52 53 54 55 56 57)(58 59 60 61 62 63 64)
         (65 66 67 68 69 70 71)(72 73 74 75 76 77 78)(79 80 81 82 83 84 85)
         (86 87 88 89 90 91 92)(93 94 95 96 97 98 99)
    ,
       (5 66 49 59 61)(10 78 88 29 12)
         (1 74 83 21 36 77 44 80 64 2 34 75 48 17 100)
         (3 15 31 52 19 11 73 79 26 56 41 99 39 84 90)
         (4 57 86 63 85 95 82 97 98 81 8 69 38 43 58)
         (6 68 89 94 92 20 13 54 24 51 87 27 76 23 67)
         (7 72 22 35 30 70 47 62 45 46 40 28 65 93 42)
         (9 71 37 91 18 55 96 60 16 53 50 25 32 14 33)
     >
                                          Type: PermutationGroup(Integer)
order(j2)


   (12)  604800
                                                    Type: PositiveInteger
(13) ->

where:


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