# 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′
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)>