Mod-3-Cohomology of MathieuGroup(12), a group of order 95040

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • MathieuGroup(12) is a group of order 95040.
  • The group order factors as 26 · 33 · 5 · 11.
  • The group is defined by Group([(1,2,3,4,5,6,7,8,9,10,11),(3,7,11,8)(4,10,5,6),(1,12)(2,11)(3,6)(4,8)(5,9)(7,10)]).
  • It is non-abelian.
  • It has 3-Rank 2.
  • The centre of a Sylow 3-subgroup has rank 1.
  • Its Sylow 3-subgroup has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

The computation was based on 3 stability conditions for H*(SmallGroup(108,17); GF(3)).

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    1  −  2·t  +  3·t2  −  3·t3  +  5·t4  −  6·t5  +  7·t6  −  7·t7  +  9·t8  −  9·t9  +  12·t10  −  11·t11  +  12·t12  −  11·t13  +  12·t14  −  9·t15  +  9·t16  −  7·t17  +  7·t18  −  6·t19  +  5·t20  −  3·t21  +  3·t22  −  2·t23  +  t24
    ( − 1  +  t)2 · (1  −  t  +  t2) · (1  +  t  +  t2) · (1  +  t2)2 · (1  −  t2  +  t4) · (1  +  t4) · (1  +  t8)
  • The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 16 minimal generators of maximal degree 16:

  1. a_3_0, a nilpotent element of degree 3
  2. a_4_1, a nilpotent element of degree 4
  3. b_4_0, an element of degree 4
  4. a_5_0, a nilpotent element of degree 5
  5. a_9_0, a nilpotent element of degree 9
  6. a_10_2, a nilpotent element of degree 10
  7. a_10_1, a nilpotent element of degree 10
  8. b_10_0, an element of degree 10
  9. a_11_2, a nilpotent element of degree 11
  10. a_11_1, a nilpotent element of degree 11
  11. a_11_0, a nilpotent element of degree 11
  12. c_12_0, a Duflot element of degree 12
  13. a_15_2, a nilpotent element of degree 15
  14. a_15_0, a nilpotent element of degree 15
  15. b_16_3, an element of degree 16
  16. b_16_1, an element of degree 16

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There are 8 "obvious" relations:
   a_3_02, a_5_02, a_9_02, a_11_02, a_11_12, a_11_22, a_15_02, a_15_22

Apart from that, there are 96 minimal relations of maximal degree 32:

  1. a_4_1·a_3_0
  2. a_4_12
  3. a_4_1·b_4_0 + a_3_0·a_5_0
  4. a_4_1·a_5_0
  5. a_3_0·a_9_0
  6. a_4_1·a_9_0
  7. a_10_1·a_3_0
  8. a_10_2·a_3_0
  9. b_10_0·a_3_0 + b_4_0·a_9_0
  10. a_4_1·a_10_1
  11. a_4_1·a_10_2
  12. a_3_0·a_11_0
  13. a_3_0·a_11_2
  14. a_5_0·a_9_0 − a_3_0·a_11_1
  15. a_4_1·b_10_0 + a_3_0·a_11_1
  16. b_4_0·a_10_1 + a_3_0·a_11_1
  17. b_4_0·a_10_2
  18. a_4_1·a_11_0
  19. a_4_1·a_11_1
  20. a_4_1·a_11_2
  21. a_10_1·a_5_0
  22. a_10_2·a_5_0
  23. b_4_0·a_11_0
  24. b_4_0·a_11_2 + b_4_03·a_3_0
  25. b_10_0·a_5_0 − b_4_0·a_11_1
  26. a_5_0·a_11_0
  27. a_5_0·a_11_1
  28. a_5_0·a_11_2 − b_4_02·a_3_0·a_5_0
  29. a_3_0·a_15_0
  30. a_3_0·a_15_2
  31. a_4_1·a_15_0
  32. a_4_1·a_15_2
  33. a_10_1·a_9_0
  34. a_10_2·a_9_0
  35. b_4_0·a_15_0 − b_4_04·a_3_0 − b_4_0·c_12_0·a_3_0
  36. b_4_0·a_15_2 − b_4_04·a_3_0 − b_4_0·c_12_0·a_3_0
  37. b_10_0·a_9_0 − b_4_0·c_12_0·a_3_0
  38. b_16_1·a_3_0 − b_4_04·a_3_0 + b_4_0·c_12_0·a_3_0
  39. b_16_3·a_3_0 − b_4_0·c_12_0·a_3_0
  40. a_10_12
  41. a_10_1·a_10_2
  42. a_10_22
  43. a_5_0·a_15_0 + b_4_03·a_3_0·a_5_0 + c_12_0·a_3_0·a_5_0
  44. a_5_0·a_15_2 + b_4_03·a_3_0·a_5_0 + c_12_0·a_3_0·a_5_0
  45. a_9_0·a_11_0
  46. a_9_0·a_11_1 − c_12_0·a_3_0·a_5_0
  47. a_9_0·a_11_2
  48. a_4_1·b_16_1 + b_4_03·a_3_0·a_5_0 − c_12_0·a_3_0·a_5_0
  49. a_4_1·b_16_3 + c_12_0·a_3_0·a_5_0
  50. a_10_1·b_10_0 − c_12_0·a_3_0·a_5_0
  51. a_10_2·b_10_0
  52. b_4_0·b_16_1 − b_4_05 + b_4_02·c_12_0
  53. b_4_0·b_16_3 − b_4_02·c_12_0
  54. b_10_02 + b_4_02·c_12_0
  55. a_10_1·a_11_0
  56. a_10_1·a_11_1
  57. a_10_1·a_11_2
  58. a_10_2·a_11_0
  59. a_10_2·a_11_1
  60. a_10_2·a_11_2
  61. b_10_0·a_11_0
  62. b_10_0·a_11_1 + b_4_0·c_12_0·a_5_0
  63. b_10_0·a_11_2 − b_4_03·a_9_0
  64. b_16_1·a_5_0 − b_4_04·a_5_0 + b_4_0·c_12_0·a_5_0
  65. b_16_3·a_5_0 − b_4_0·c_12_0·a_5_0
  66. a_11_0·a_11_1
  67. a_11_0·a_11_2
  68. a_11_1·a_11_2 − b_4_02·a_3_0·a_11_1
  69. a_9_0·a_15_0
  70. a_9_0·a_15_2
  71. a_10_1·a_15_0
  72. a_10_1·a_15_2
  73. a_10_2·a_15_0
  74. a_10_2·a_15_2
  75. b_10_0·a_15_0 + b_4_04·a_9_0 + b_4_0·c_12_0·a_9_0
  76. b_10_0·a_15_2 + b_4_04·a_9_0 + b_4_0·c_12_0·a_9_0
  77. b_16_1·a_9_0 − b_4_04·a_9_0 + b_4_0·c_12_0·a_9_0
  78. b_16_3·a_9_0 − b_4_0·c_12_0·a_9_0
  79. a_11_0·a_15_2
  80. a_11_1·a_15_0 + a_11_0·a_15_0 + b_4_03·a_3_0·a_11_1 + c_12_0·a_3_0·a_11_1
  81. a_11_1·a_15_2 + b_4_03·a_3_0·a_11_1 + c_12_0·a_3_0·a_11_1
  82. a_11_2·a_15_2 − a_11_2·a_15_0 − a_11_0·a_15_0
  83. a_10_1·b_16_1 + b_4_03·a_3_0·a_11_1 − c_12_0·a_3_0·a_11_1
  84. a_10_1·b_16_3 + a_11_0·a_15_0 + c_12_0·a_3_0·a_11_1
  85. a_10_2·b_16_1 + a_11_2·a_15_0 + a_11_0·a_15_0
  86. a_10_2·b_16_3 + a_11_0·a_15_0
  87. b_10_0·b_16_1 − b_4_04·b_10_0 + b_4_0·b_10_0·c_12_0
  88. b_10_0·b_16_3 − b_4_0·b_10_0·c_12_0
  89. b_16_1·a_11_0
  90. b_16_1·a_11_1 − b_4_04·a_11_1 + b_4_0·c_12_0·a_11_1
  91. b_16_3·a_11_1 + b_16_3·a_11_0 − b_4_0·c_12_0·a_11_1
  92. b_16_3·a_11_2 + b_16_3·a_11_0 + b_4_03·c_12_0·a_3_0
  93. a_15_0·a_15_2
  94. b_16_1·a_15_2 − b_16_1·a_15_0
  95. b_16_3·a_15_2 − b_4_04·c_12_0·a_3_0 − b_4_0·c_12_02·a_3_0
  96. b_16_1·b_16_3 − b_4_05·c_12_0 + b_4_02·c_12_02

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 32 using the Hilbert-Poincaré criterion.
  • The completion test was perfect: It applied in the last degree in which a generator or relation was found.
  • The following is a filter regular homogeneous system of parameters:
    1. c_12_0, an element of degree 12
    2. b_16_3 + b_16_1, an element of degree 16
  • A Duflot regular sequence is given by c_12_0.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 26].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(108,17); GF(3))

  1. a_3_0 → a_3_2 − a_3_1 − a_3_0
  2. a_4_1 → a_4_3
  3. b_4_0 → b_4_1 + b_4_0
  4. a_5_0 → a_5_0
  5. a_9_0 → a_9_1
  6. a_10_2 → a_10_0
  7. a_10_1 → a_10_2
  8. b_10_0 → b_10_1
  9. a_11_2 → a_11_1 + b_4_02·a_3_0
  10. a_11_1 → a_11_4
  11. a_11_0 → a_11_5
  12. c_12_0 → b_4_23 − c_12_4
  13. a_15_2 → b_4_0·a_11_1 − b_4_03·a_3_0 + c_12_4·a_3_0
  14. a_15_0 → b_4_2·a_11_1 − b_4_03·a_3_0 − c_12_4·a_3_1 + c_12_4·a_3_0
  15. b_16_3 → b_4_2·c_12_4 − b_4_0·c_12_4
  16. b_16_1 → b_4_04 + b_4_0·c_12_4

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

  1. a_3_0 → 0, an element of degree 3
  2. a_4_1 → 0, an element of degree 4
  3. b_4_0 → 0, an element of degree 4
  4. a_5_0 → 0, an element of degree 5
  5. a_9_0 → 0, an element of degree 9
  6. a_10_2 → 0, an element of degree 10
  7. a_10_1 → 0, an element of degree 10
  8. b_10_0 → 0, an element of degree 10
  9. a_11_2 → 0, an element of degree 11
  10. a_11_1 → 0, an element of degree 11
  11. a_11_0 → 0, an element of degree 11
  12. c_12_0 →  − c_2_06, an element of degree 12
  13. a_15_2 → 0, an element of degree 15
  14. a_15_0 → 0, an element of degree 15
  15. b_16_3 → 0, an element of degree 16
  16. b_16_1 → 0, an element of degree 16

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. a_3_0 → 0, an element of degree 3
  2. a_4_1 → 0, an element of degree 4
  3. b_4_0 → 0, an element of degree 4
  4. a_5_0 → 0, an element of degree 5
  5. a_9_0 → 0, an element of degree 9
  6. a_10_2 → c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  7. a_10_1 → c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  8. b_10_0 → 0, an element of degree 10
  9. a_11_2 →  − c_2_1·c_2_24·a_1_0 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0
       − c_2_14·c_2_2·a_1_1, an element of degree 11
  10. a_11_1 →  − c_2_1·c_2_24·a_1_0 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0
       − c_2_14·c_2_2·a_1_1, an element of degree 11
  11. a_11_0 → c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, an element of degree 11
  12. c_12_0 →  − c_2_26 − c_2_12·c_2_24 − c_2_14·c_2_22 − c_2_16, an element of degree 12
  13. a_15_2 → 0, an element of degree 15
  14. a_15_0 → c_2_1·c_2_26·a_1_0 − c_2_13·c_2_24·a_1_0 − c_2_14·c_2_23·a_1_1
       + c_2_16·c_2_2·a_1_1, an element of degree 15
  15. b_16_3 →  − c_2_12·c_2_26 − c_2_14·c_2_24 − c_2_16·c_2_22, an element of degree 16
  16. b_16_1 → 0, an element of degree 16

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. a_3_0 →  − c_2_2·a_1_1, an element of degree 3
  2. a_4_1 → c_2_2·a_1_0·a_1_1, an element of degree 4
  3. b_4_0 → c_2_22, an element of degree 4
  4. a_5_0 →  − c_2_22·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
  5. a_9_0 → c_2_1·c_2_23·a_1_1 − c_2_13·c_2_2·a_1_1, an element of degree 9
  6. a_10_2 → 0, an element of degree 10
  7. a_10_1 → c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  8. b_10_0 → c_2_1·c_2_24 − c_2_13·c_2_22, an element of degree 10
  9. a_11_2 → c_2_25·a_1_1, an element of degree 11
  10. a_11_1 →  − c_2_1·c_2_24·a_1_0 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0
       − c_2_14·c_2_2·a_1_1, an element of degree 11
  11. a_11_0 → 0, an element of degree 11
  12. c_12_0 →  − c_2_12·c_2_24 − c_2_14·c_2_22 − c_2_16, an element of degree 12
  13. a_15_2 →  − c_2_27·a_1_1 + c_2_12·c_2_25·a_1_1 + c_2_14·c_2_23·a_1_1 + c_2_16·c_2_2·a_1_1, an element of degree 15
  14. a_15_0 →  − c_2_27·a_1_1 + c_2_12·c_2_25·a_1_1 + c_2_14·c_2_23·a_1_1 + c_2_16·c_2_2·a_1_1, an element of degree 15
  15. b_16_3 →  − c_2_12·c_2_26 − c_2_14·c_2_24 − c_2_16·c_2_22, an element of degree 16
  16. b_16_1 → c_2_28 + c_2_12·c_2_26 + c_2_14·c_2_24 + c_2_16·c_2_22, an element of degree 16

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. a_3_0 →  − c_2_2·a_1_1, an element of degree 3
  2. a_4_1 →  − c_2_2·a_1_0·a_1_1, an element of degree 4
  3. b_4_0 → c_2_22, an element of degree 4
  4. a_5_0 → c_2_22·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
  5. a_9_0 →  − c_2_1·c_2_23·a_1_1 + c_2_13·c_2_2·a_1_1, an element of degree 9
  6. a_10_2 → 0, an element of degree 10
  7. a_10_1 → c_2_1·c_2_23·a_1_0·a_1_1 − c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  8. b_10_0 →  − c_2_1·c_2_24 + c_2_13·c_2_22, an element of degree 10
  9. a_11_2 → c_2_25·a_1_1, an element of degree 11
  10. a_11_1 →  − c_2_1·c_2_24·a_1_0 + c_2_12·c_2_23·a_1_1 + c_2_13·c_2_22·a_1_0
       − c_2_14·c_2_2·a_1_1, an element of degree 11
  11. a_11_0 → 0, an element of degree 11
  12. c_12_0 →  − c_2_12·c_2_24 − c_2_14·c_2_22 − c_2_16, an element of degree 12
  13. a_15_2 →  − c_2_27·a_1_1 + c_2_12·c_2_25·a_1_1 + c_2_14·c_2_23·a_1_1 + c_2_16·c_2_2·a_1_1, an element of degree 15
  14. a_15_0 →  − c_2_27·a_1_1 + c_2_12·c_2_25·a_1_1 + c_2_14·c_2_23·a_1_1 + c_2_16·c_2_2·a_1_1, an element of degree 15
  15. b_16_3 →  − c_2_12·c_2_26 − c_2_14·c_2_24 − c_2_16·c_2_22, an element of degree 16
  16. b_16_1 → c_2_28 + c_2_12·c_2_26 + c_2_14·c_2_24 + c_2_16·c_2_22, an element of degree 16

Restriction map to a maximal el. ab. subgp. of rank 2 in a Sylow subgroup

  1. a_3_0 → 0, an element of degree 3
  2. a_4_1 → 0, an element of degree 4
  3. b_4_0 → 0, an element of degree 4
  4. a_5_0 → 0, an element of degree 5
  5. a_9_0 → 0, an element of degree 9
  6. a_10_2 →  − c_2_1·c_2_23·a_1_0·a_1_1 + c_2_13·c_2_2·a_1_0·a_1_1, an element of degree 10
  7. a_10_1 → 0, an element of degree 10
  8. b_10_0 → 0, an element of degree 10
  9. a_11_2 → c_2_1·c_2_24·a_1_0 − c_2_12·c_2_23·a_1_1 − c_2_13·c_2_22·a_1_0
       + c_2_14·c_2_2·a_1_1, an element of degree 11
  10. a_11_1 → 0, an element of degree 11
  11. a_11_0 → 0, an element of degree 11
  12. c_12_0 →  − c_2_26 − c_2_12·c_2_24 − c_2_14·c_2_22 − c_2_16, an element of degree 12
  13. a_15_2 → c_2_1·c_2_26·a_1_0 − c_2_13·c_2_24·a_1_0 − c_2_14·c_2_23·a_1_1
       + c_2_16·c_2_2·a_1_1, an element of degree 15
  14. a_15_0 → c_2_1·c_2_26·a_1_0 − c_2_13·c_2_24·a_1_0 − c_2_14·c_2_23·a_1_1
       + c_2_16·c_2_2·a_1_1, an element of degree 15
  15. b_16_3 → 0, an element of degree 16
  16. b_16_1 → c_2_12·c_2_26 + c_2_14·c_2_24 + c_2_16·c_2_22, an element of degree 16

About the group Ring generators Ring relations Completion information Restriction maps


Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2

Last change: 14.12.2010