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:
- The points of the permutation are numbered 1..n
- numbers in brackets are points of permutations represented in cyclic notation.
- The permutation is represented by a set of comma seperated permutations in angle brackets like this: <(1 2)(3 4),(1 2 3)>
- non-changing elements of the permutation are ommited so the above case is equivalent to: <(1 2)(3 4),(1 2 3)(4)>
