CoCalc -- Mathieu

Project: KOB1

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︠553727c3-9ad0-4edf-99c5-a9c49678f33d︠

︠2a31cbdd-a4f0-48c2-8e57-87a28aba45adi︠
︠076af10e-789e-4a0d-b148-feb51011e756︠
G = MathieuGroup(24)
G
︠3159b0d9-4863-4879-bb29-e5972ef0bf2ei︠
%md
We define $S$ to be a complete set of representatives of the conjugacy classes of subgroups of $M_{24}$.

We define $S$ to be a complete set of representatives of the conjugacy classes of subgroups of $M_{24}$ .

S = G.conjugacy_classes_subgroups()
len(S)

1529

Then we pick out the subgroups from $S$ of order 24.

S24 = [H for H in S if H.order() == 24]
len(S24)

51

So there are 51 conjugacy classes of subgroups of $M_{24}$ of order 24. We list the generators for each representative.

Note that the default Sage representatiion for permutations is that products of cycles are evaluated from left-to-right, i.e. the opposite of the usual right-to-left compositions of functions convention.

j=1
for H in S24:
    print('Conjugacy class %i'%j)
    for h in H.gens():
        h
    j=j+1

Conjugacy class 1
(1,4,6,19)(2,12,16,11)(3,24)(5,7)(8,13)(9,14)(10,22,23,15)(17,20,18,21)
(1,4,19,6)(2,16,12,11)(3,14)(5,9)(7,24)(8,13)(10,22,15,23)(17,21,20,18)
Conjugacy class 2
(1,19,11)(2,21,4)(3,13,17)(5,16,8)(6,9,7)(10,12,20)(14,24,23)(15,22,18)
(1,19,7,6)(2,21,24,23)(3,20)(4,14)(5,16,18,15)(8,22)(9,11)(10,12,13,17)
Conjugacy class 3
(3,14,10)(5,7,22)(8,9,18)(12,19,16)(13,24,20)(15,23,21)
(1,9)(2,14)(3,21)(4,13)(5,24)(6,15)(7,20)(8,12)(10,23)(11,22)(16,18)(17,19)
Conjugacy class 4
(1,13,5)(3,8,11)(6,23,12)(7,24,10)(9,14,20)(16,19,17)
(1,18)(2,10)(4,20)(6,17)(11,22)(12,15)(16,23)(19,21)
Conjugacy class 5
(1,10,13)(3,20,9)(4,22,8)(5,18,24)(6,17,12)(15,16,21)
(1,18)(4,20)(6,21)(8,9)(12,16)(13,24)(15,23)(17,19)
Conjugacy class 6
(1,6)(2,23)(3,8)(4,9)(5,16)(7,19)(10,12)(11,14)(13,17)(15,18)(20,22)(21,24)
(1,24)(2,7)(3,23,20,21)(4,17,14,12)(5,13)(6,22,19,8)(9,15,11,16)(10,18)
Conjugacy class 7
(1,10,13)(3,20,9)(4,22,8)(5,18,24)(6,17,12)(15,16,21)
(1,18)(2,7)(3,19)(4,21)(5,10)(6,9)(8,16)(11,17)(12,20)(13,24)(14,15)(22,23)
Conjugacy class 8
(1,13)(2,18)(4,14)(5,24)(6,19)(7,10)(8,22)(12,17)
(1,19,9,7,12,20)(2,21,14,24,15,22)(3,10,17,11,13,6)(4,5,23,8,18,16)
Conjugacy class 9
(1,13,5)(3,8,11)(6,23,12)(7,24,10)(9,14,20)(16,19,17)
(1,13,10,7)(2,18)(3,4,11,14)(6,19,17,12)(8,9,22,20)(15,21)
Conjugacy class 10
(1,10)(2,24)(3,9)(4,14)(5,18)(6,19)(7,13)(8,22)(11,20)(12,17)(15,21)(16,23)
(1,24,5,13,18,2)(4,14,20)(6,17,16,12,19,21)(7,10)(8,22,9)(15,23)
Conjugacy class 11
(2,7,18)(4,9,22)(5,10,24)(6,15,21)(8,20,14)(12,23,16)
(1,2,10,13,5,7)(4,22,9)(6,19,21,12,17,16)(8,14,20)(15,23)(18,24)
Conjugacy class 12
(3,11)(4,8)(6,23)(7,10)(12,15)(16,19)(17,21)(18,24)
(1,5,18)(2,24,13)(4,20,14)(6,16,19)(8,9,22)(12,21,17)
Conjugacy class 13
(1,7)(2,10)(3,23)(4,12)(5,18)(6,11)(8,15)(9,19)(13,24)(14,17)(16,22)(20,21)
(1,13,2)(4,8,11)(9,22,20)(10,18,24)(12,15,23)(17,21,19)
Conjugacy class 14
(1,13,18,10,7,2)(3,8,4,9,14,22)(5,24)(6,19,16)(11,20)(12,23,17)
(1,13)(2,18)(4,14)(5,24)(6,19)(7,10)(8,22)(12,17)
(1,18,7)(2,13,10)(3,22,14,9,4,8)(6,23,19,17,16,12)(11,20)(15,21)
Conjugacy class 15
(1,3)(2,4)(5,8)(6,19)(7,20)(9,10)(11,13)(12,17)(14,24)(15,21)(16,23)(18,22)
(1,5)(2,13)(3,6)(4,15)(7,24)(8,23)(9,19)(10,18)(11,12)(14,21)(16,22)(17,20)
(1,18)(2,7)(3,19)(4,21)(5,10)(6,9)(8,16)(11,17)(12,20)(13,24)(14,15)(22,23)
Conjugacy class 16
(1,2)(3,8)(4,11)(5,13)(6,17)(7,20)(9,10)(12,21)(14,18)(15,16)(19,23)(22,24)
(1,5)(2,16)(3,9)(4,13)(6,22)(7,11)(8,12)(10,17)(14,21)(15,19)(18,24)(20,23)
(1,19)(2,4)(3,14)(7,15)(8,17)(10,24)(13,20)(21,22)
Conjugacy class 17
(3,11)(4,8)(6,23)(7,10)(12,15)(16,19)(17,21)(18,24)
(1,9)(2,3)(4,13)(5,22)(6,15)(7,11)(8,18)(10,14)(12,17)(16,19)(20,24)(21,23)
(1,22)(2,8)(3,24)(4,10)(5,9)(6,17)(7,20)(11,13)(12,23)(14,18)(15,21)(16,19)
Conjugacy class 18
(1,3)(2,5)(4,8)(6,7)(9,19)(10,15)(11,13)(12,14)(16,24)(17,18)(20,21)(22,23)
(1,10,17,3)(2,24,16,14)(4,15,18,8)(5,11,13,12)(6,22,20,7)(9,19,23,21)
Conjugacy class 19
(2,11)(3,24,8,14)(5,9,13,7)(10,18,22,20)(12,16)(15,21,23,17)
(1,14,19,24)(2,21,16,17)(4,13,6,5)(10,15)(11,18,12,20)(22,23)
Conjugacy class 20
(1,2,10,13,5,7)(4,22,9)(6,19,21,12,17,16)(8,14,20)(15,23)(18,24)
(1,24,5,13,18,2)(4,14,20)(6,17,16,12,19,21)(7,10)(8,22,9)(15,23)
Conjugacy class 21
(1,5)(2,16)(3,9)(4,13)(6,22)(7,11)(8,12)(10,17)(14,21)(15,19)(18,24)(20,23)
(1,21,12,4,14,2)(3,23,24,7,10,18)(5,6,8,13,22,16)(9,20,15,11,17,19)
Conjugacy class 22
(1,13,5)(3,8,11)(6,23,12)(7,24,10)(9,14,20)(16,19,17)
(1,19)(2,15)(3,11)(4,14)(5,23)(6,7)(8,22)(9,20)(10,12)(13,17)(16,24)(18,21)
Conjugacy class 23
(1,4,18,9,24,8)(2,3,5,14,10,22)(6,16,12)(7,11)(13,20)(17,23,19)
(1,18)(4,20)(6,21)(8,9)(12,16)(13,24)(15,23)(17,19)
Conjugacy class 24
(1,5)(2,13)(6,12)(9,22)(14,20)(15,23)(16,17)(19,21)
(1,14,23,5,22,6)(2,20,15,13,9,12)(3,17,7,11,16,18)(4,19,24,8,21,10)
Conjugacy class 25
(1,13,10,6)(2,9,21,4)(3,16,24,11)(5,23,22,8)(7,17)(15,18)
(1,20)(2,17)(5,8)(6,13)(7,21)(10,14)(11,24)(15,18)
Conjugacy class 26
(1,7,2)(4,14,11)(6,19,23)(8,20,22)(10,13,18)(12,16,17)
(1,19)(2,15)(3,11)(4,14)(5,23)(6,7)(8,22)(9,20)(10,12)(13,17)(16,24)(18,21)
Conjugacy class 27
(1,14)(3,12,20,11)(6,24,19,13)(7,22,18,23)(8,9,21,17)(10,16)
(1,16,10)(2,5,15)(3,13,19)(6,20,24)(8,23,9)(17,21,22)
Conjugacy class 28
(1,9)(2,3)(4,13)(5,22)(6,15)(7,11)(8,18)(10,14)(12,17)(16,19)(20,24)(21,23)
(1,9)(2,8,13,11)(3,24)(4,10)(5,22)(6,16,21,23)(7,14,18,20)(12,15,19,17)
Conjugacy class 29
(1,4)(3,8)(6,19)(7,9)(10,15)(17,18)(20,21)(22,23)
(1,17,8,19,20,3)(2,12,16)(4,21,9,6,18,7)(5,24)(13,14)(15,23,22)
Conjugacy class 30
(1,10,13)(3,20,9)(4,22,8)(5,18,24)(6,17,12)(15,16,21)
(1,16)(2,12)(3,11)(4,22)(5,6)(7,15)(8,14)(9,20)(10,23)(13,21)(17,24)(18,19)
Conjugacy class 31
(1,14,23,24,11,19)(2,8,6,5,4,12)(3,17,13,22,15,18)(7,20,21,10,9,16)
(1,18)(4,20)(6,21)(8,9)(12,16)(13,24)(15,23)(17,19)
Conjugacy class 32
(1,8,16,5,20,23)(2,3,17,13,22,6)(4,15,7,9,21,18)(10,11,12,24,14,19)
(1,21,9)(2,19,14)(3,24,17)(4,5,15)(6,22,10)(7,16,8)(11,13,12)(18,23,20)
Conjugacy class 33
(1,2)(3,9)(4,20)(6,23)(8,14)(10,18)(11,22)(16,17)
(1,13,5)(3,8,11)(6,23,12)(7,24,10)(9,14,20)(16,19,17)
Conjugacy class 34
(1,13,5)(3,8,11)(6,23,12)(7,24,10)(9,14,20)(16,19,17)
(1,13,2,10,7,18)(3,9)(4,8,11,22,14,20)(5,24)(6,19,23)(12,16,17)
Conjugacy class 35
(1,7,10,18,5,2)(3,22,4)(6,23,21,16,17,12)(11,20,14)(13,24)(15,19)
(1,13,7,18,24,5)(2,10)(3,8,22)(6,16)(9,11,14)(12,23,15,21,17,19)
Conjugacy class 36
(1,18)(2,7)(3,19)(4,21)(5,10)(6,9)(8,16)(11,17)(12,20)(13,24)(14,15)(22,23)
(1,19,10,17)(2,23,18,15)(3,22,20,8)(4,9,14,11)(5,16,24,21)(6,13,12,7)
Conjugacy class 37
(1,8)(2,11)(3,5)(4,13)(6,16)(7,14)(9,18)(10,22)(12,21)(15,17)(19,23)(20,24)
(1,24,10,2)(3,19,9,6)(4,15,14,21)(5,13,18,7)(8,23,22,16)(11,17,20,12)
Conjugacy class 38
(1,3,23,13,20,16)(2,8,6,5,22,17)(4,19,24,14,12,18)(7,9,15,10,11,21)
(1,19,9,13,12,11)(2,21,14,5,15,4)(3,10,6,20,7,17)(8,18,23,22,24,16)
Conjugacy class 39
(1,11)(2,4)(3,5)(6,16)(7,14)(8,13)(9,24)(10,20)(12,19)(15,21)(17,23)(18,22)
(1,12,14)(2,17,11)(3,18,21)(4,24,15)(5,6,8)(7,16,9)(10,23,22)(13,19,20)
Conjugacy class 40
(1,5)(2,18)(3,16)(4,19)(6,11)(7,13)(8,15)(9,23)(10,24)(12,22)(14,21)(17,20)
(1,16,3)(2,12,20)(4,5,19)(6,11,10)(7,17,22)(8,24,15)(9,18,21)(13,23,14)
Conjugacy class 41
(2,9)(4,16)(6,23)(7,15)(8,19)(10,12)(13,14)(18,24)
(2,14,15)(3,12,20)(6,13,22)(7,19,21)(8,23,9)(17,24,18)
Conjugacy class 42
(1,4,2,11)(3,24,8,7)(5,14,13,9)(6,16)(10,22,18,20)(17,23)
(1,24)(2,7)(3,20)(4,11)(6,21)(8,22)(9,14)(19,23)
Conjugacy class 43
(1,2,10)(3,11)(4,14,20,8,22,9)(5,7,13)(6,15,17,12,23,19)(16,21)
(1,12)(2,19)(3,11)(4,9)(5,17)(6,13)(7,23)(8,20)(10,15)(14,22)(16,24)(18,21)
Conjugacy class 44
(1,8)(2,20)(3,9)(4,6)(5,10)(7,19)(11,22)(12,18)(13,17)(14,23)(15,16)(21,24)
(1,16,11,3,14,12)(2,8,7,23,5,22)(4,18,17,9,21,15)(6,20,24,10,13,19)
Conjugacy class 45
(1,5)(2,13)(6,12)(9,22)(14,20)(15,23)(16,17)(19,21)
(1,20,18,9,5,8,2,22,10,14,13,3)(4,24,11,7)(6,12,17,23,15,16)(19,21)
Conjugacy class 46
(1,8)(2,11)(3,5)(4,13)(6,16)(7,14)(9,18)(10,22)(12,21)(15,17)(19,23)(20,24)
(1,16)(2,12)(3,11)(4,22)(5,6)(7,15)(8,14)(9,20)(10,23)(13,21)(17,24)(18,19)
Conjugacy class 47
(1,13)(2,18)(4,14)(5,24)(6,19)(7,10)(8,22)(12,17)
(1,17,24,16,5,12,13,19,18,21,2,6)(3,11)(4,9,14,8,20,22)(7,23,10,15)
Conjugacy class 48
(1,17,14,8,20,7,12,15,11,18,24,9)(2,16,4,19,22,10,21,13,5,3,23,6)
(1,20)(2,5)(3,10)(4,21)(6,19)(8,15)(12,24)(17,18)
Conjugacy class 49
(1,4)(2,20,13,11,18,8)(3,10,9,5,22,24)(6,15,21)(7,14)(16,23,19)
(1,7)(2,5,13,24,18,10)(3,22,9)(6,16,21,19,15,23)(8,11,20)(12,17)
Conjugacy class 50
(1,12,20,24,6,14,2,21,22,13,23,3)(4,7,16,9,10,19,11,5,17,8,18,15)
(1,17,22,10,6,4)(2,16,20,18,23,11)(3,7,21,8,24,19)(5,12,9,13,15,14)
Conjugacy class 51
(1,9,3,23)(2,10,18,20)(4,13,15,14)(5,11,17,16)(6,21,24,8)(7,12,22,19)
(1,22)(2,6)(3,21)(4,10)(5,12)(7,14)(8,13)(9,16)(11,15)(17,24)(18,19)(20,23)

Error in lines 1-1
Traceback (most recent call last):
  File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute
    exec compile(block+'\n', '', 'single') in namespace, locals
  File "", line 1, in <module>
NameError: name 'H' is not defined