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Mod-2-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 2-Rank 3.
• The centre of a Sylow 2-subgroup has rank 1.
• Its Sylow 2-subgroup has 5 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 3.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  ( − 1)·((1  −  t  +  t2)2)

  ( − 1  +  t)3 · (1  +  t2) · (1  +  t  +  t2)

• The a-invariants are -∞,-∞,-∞,-3. 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, -3, -3].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 8 minimal generators of maximal degree 7:

1. b_2_0, an element of degree 2
2. b_3_2, an element of degree 3
3. b_3_1, an element of degree 3
4. b_3_0, an element of degree 3
5. c_4_0, a Duflot element of degree 4
6. b_5_0, an element of degree 5
7. b_6_3, an element of degree 6
8. b_7_1, an element of degree 7

About the group

Ring generators

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Ring relations

There are 14 minimal relations of maximal degree 14:

1. b_2_0·b_3_2
2. b_3_0·b_3_2
3. b_3_1² + b_3_0·b_3_1 + b_3_0² + b_2_0³
4. b_3_1·b_3_2
5. b_2_0·b_5_0
6. b_3_0·b_5_0
7. b_3_1·b_5_0
8. b_6_3·b_3_0
9. b_6_3·b_3_1 + b_2_0·b_7_1
10. b_3_0·b_7_1
11. b_3_1·b_7_1 + b_2_0²·b_6_3
12. b_5_0² + b_3_2·b_7_1
13. b_5_0·b_7_1 + b_6_3·b_3_2² + c_4_0·b_3_2·b_5_0
14. b_7_1² + b_6_3·b_3_2·b_5_0 + b_2_0·b_6_3² + c_4_0·b_3_2·b_7_1

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 14 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. b_2_0² + c_4_0, an element of degree 4
  2. b_3_2² + b_3_0² + b_6_3 + b_2_0·c_4_0, an element of degree 6
  3. b_3_0² + b_6_3, an element of degree 6
• A Duflot regular sequence is given by c_4_0.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 13].
• Modifying the above filter regular HSOP, we obtained the following parameters:
1. b_2_0² + c_4_0, an element of degree 4
2. b_3_2 + b_3_1, an element of degree 3
3. b_3_0² + b_6_3, an element of degree 6

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(192,1494); GF(2))

1. b_2_0 → b_2_0
2. b_3_2 → b_3_0
3. b_3_1 → b_3_1
4. b_3_0 → b_3_2
5. c_4_0 → b_1_0·b_3_0 + b_1_0⁴ + b_2_2² + c_4_6
6. b_5_0 → b_2_2·b_3_0 + c_4_6·b_1_0
7. b_6_3 → b_2_2²·b_1_0² + b_2_2·c_4_6
8. b_7_1 → b_2_2²·b_3_0 + c_4_6·b_3_4 + c_4_6·b_3_0 + c_4_6·b_1_0³ + b_2_2·c_4_6·b_1_0

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

1. b_2_0 → 0, an element of degree 2
2. b_3_2 → 0, an element of degree 3
3. b_3_1 → 0, an element of degree 3
4. b_3_0 → 0, an element of degree 3
5. c_4_0 → c_1_0⁴, an element of degree 4
6. b_5_0 → 0, an element of degree 5
7. b_6_3 → 0, an element of degree 6
8. b_7_1 → 0, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

1. b_2_0 → 0, an element of degree 2
2. b_3_2 → c_1_0·c_1_1² + c_1_0²·c_1_1, an element of degree 3
3. b_3_1 → 0, an element of degree 3
4. b_3_0 → 0, an element of degree 3
5. c_4_0 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2
      + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
6. b_5_0 → c_1_0·c_1_1⁴ + c_1_0⁴·c_1_1, an element of degree 5
7. b_6_3 → c_1_1²·c_1_2⁴ + c_1_1⁴·c_1_2² + c_1_0·c_1_1·c_1_2⁴ + c_1_0·c_1_1⁴·c_1_2
      + c_1_0²·c_1_2⁴ + c_1_0²·c_1_1²·c_1_2² + c_1_0⁴·c_1_2² + c_1_0⁴·c_1_1·c_1_2, an element of degree 6
8. b_7_1 → c_1_0·c_1_1⁶ + c_1_0²·c_1_1⁵ + c_1_0³·c_1_1⁴ + c_1_0⁴·c_1_1³
      + c_1_0⁵·c_1_1² + c_1_0⁶·c_1_1, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

1. b_2_0 → c_1_2² + c_1_1·c_1_2 + c_1_1², an element of degree 2
2. b_3_2 → 0, an element of degree 3
3. b_3_1 → c_1_2³ + c_1_1²·c_1_2 + c_1_1³, an element of degree 3
4. b_3_0 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
5. c_4_0 → c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2 + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2
      + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
6. b_5_0 → 0, an element of degree 5
7. b_6_3 → 0, an element of degree 6
8. b_7_1 → 0, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

1. b_2_0 → c_1_1², an element of degree 2
2. b_3_2 → 0, an element of degree 3
3. b_3_1 → c_1_1³, an element of degree 3
4. b_3_0 → 0, an element of degree 3
5. c_4_0 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2
      + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
6. b_5_0 → 0, an element of degree 5
7. b_6_3 → c_1_0·c_1_1·c_1_2⁴ + c_1_0·c_1_1³·c_1_2² + c_1_0²·c_1_2⁴
      + c_1_0²·c_1_1³·c_1_2 + c_1_0⁴·c_1_2² + c_1_0⁴·c_1_1·c_1_2, an element of degree 6
8. b_7_1 → c_1_0·c_1_1²·c_1_2⁴ + c_1_0·c_1_1⁴·c_1_2² + c_1_0²·c_1_1·c_1_2⁴
      + c_1_0²·c_1_1⁴·c_1_2 + c_1_0⁴·c_1_1·c_1_2² + c_1_0⁴·c_1_1²·c_1_2, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

1. b_2_0 → c_1_1², an element of degree 2
2. b_3_2 → 0, an element of degree 3
3. b_3_1 → c_1_1³, an element of degree 3
4. b_3_0 → 0, an element of degree 3
5. c_4_0 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2
      + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
6. b_5_0 → 0, an element of degree 5
7. b_6_3 → c_1_0·c_1_1·c_1_2⁴ + c_1_0·c_1_1³·c_1_2² + c_1_0²·c_1_2⁴
      + c_1_0²·c_1_1³·c_1_2 + c_1_0⁴·c_1_2² + c_1_0⁴·c_1_1·c_1_2, an element of degree 6
8. b_7_1 → c_1_0·c_1_1²·c_1_2⁴ + c_1_0·c_1_1⁴·c_1_2² + c_1_0²·c_1_1·c_1_2⁴
      + c_1_0²·c_1_1⁴·c_1_2 + c_1_0⁴·c_1_1·c_1_2² + c_1_0⁴·c_1_1²·c_1_2, an element of degree 7

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

1. b_2_0 → 0, an element of degree 2
2. b_3_2 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
3. b_3_1 → 0, an element of degree 3
4. b_3_0 → 0, an element of degree 3
5. c_4_0 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2
      + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
6. b_5_0 → c_1_1·c_1_2⁴ + c_1_1⁴·c_1_2, an element of degree 5
7. b_6_3 → c_1_0·c_1_1·c_1_2⁴ + c_1_0·c_1_1⁴·c_1_2 + c_1_0²·c_1_2⁴
      + c_1_0²·c_1_1²·c_1_2² + c_1_0²·c_1_1⁴ + c_1_0⁴·c_1_2² + c_1_0⁴·c_1_1·c_1_2
      + c_1_0⁴·c_1_1², an element of degree 6
8. b_7_1 → c_1_1·c_1_2⁶ + c_1_1²·c_1_2⁵ + c_1_1³·c_1_2⁴ + c_1_1⁴·c_1_2³
      + c_1_1⁵·c_1_2² + c_1_1⁶·c_1_2, an element of degree 7

About the group

Ring generators

Ring relations

Completion information

Restriction maps