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Singular

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Mod-2-Cohomology of group number 1494 of order 192

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• The group order factors as 2⁶ · 3.
• 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  +  3·t2  −  t3  +  t4)

  ( − 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 4:

1. b_1_0, an element of degree 1
2. b_2_2, an element of degree 2
3. b_2_0, an element of degree 2
4. b_3_4, an element of degree 3
5. b_3_2, an element of degree 3
6. b_3_1, an element of degree 3
7. b_3_0, an element of degree 3
8. c_4_6, a Duflot element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 14 minimal relations of maximal degree 6:

1. b_2_0·b_1_0
2. b_1_0·b_3_1
3. b_1_0·b_3_2
4. b_1_0·b_3_4
5. b_2_0·b_3_0
6. b_2_2·b_3_1 + b_2_0·b_3_4
7. b_2_2·b_3_2
8. b_3_0·b_3_1
9. b_3_0·b_3_2
10. b_3_0·b_3_4 + b_3_0² + b_1_0³·b_3_0 + b_2_2·b_1_0·b_3_0 + c_4_6·b_1_0²
11. b_3_1·b_3_4 + b_2_0²·b_2_2
12. b_3_2² + b_3_1·b_3_2 + b_3_1² + b_2_0³
13. b_3_2·b_3_4
14. b_3_4² + b_3_0² + b_1_0³·b_3_0 + b_2_2·b_1_0·b_3_0 + b_2_0·b_2_2² + c_4_6·b_1_0²

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 6 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_4_6, an element of degree 4
  2. b_1_0² + b_2_2 + b_2_0, an element of degree 2
  3. b_3_4 + b_3_2 + b_2_2·b_1_0, an element of degree 3
• A Duflot regular sequence is given by c_4_6.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 6].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(Syl2(M12); GF(2))

1. b_1_0 → b_1_0
2. b_2_2 → b_2_5 + b_2_4
3. b_2_0 → b_1_2² + b_1_1·b_1_2 + b_1_1² + b_2_4
4. b_3_4 → b_2_5·b_1_2 + b_2_4·b_1_1
5. b_3_2 → b_1_1·b_1_2² + b_1_1²·b_1_2 + b_2_4·b_1_1
6. b_3_1 → b_1_2³ + b_1_1²·b_1_2 + b_1_1³ + b_2_4·b_1_2 + b_2_4·b_1_1
7. b_3_0 → b_3_9
8. c_4_6 → b_2_4·b_1_1·b_1_2 + b_2_4² + c_4_14

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

1. b_1_0 → 0, an element of degree 1
2. b_2_2 → 0, an element of degree 2
3. b_2_0 → 0, an element of degree 2
4. b_3_4 → 0, an element of degree 3
5. b_3_2 → 0, an element of degree 3
6. b_3_1 → 0, an element of degree 3
7. b_3_0 → 0, an element of degree 3
8. c_4_6 → c_1_0⁴, an element of degree 4

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

1. b_1_0 → c_1_1, an element of degree 1
2. b_2_2 → c_1_2² + c_1_1·c_1_2, an element of degree 2
3. b_2_0 → 0, an element of degree 2
4. b_3_4 → 0, an element of degree 3
5. b_3_2 → 0, an element of degree 3
6. b_3_1 → 0, an element of degree 3
7. b_3_0 → c_1_0·c_1_1² + c_1_0²·c_1_1, an element of degree 3
8. c_4_6 → c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2 + c_1_0·c_1_1³ + c_1_0²·c_1_2²
      + c_1_0²·c_1_1·c_1_2 + c_1_0⁴, an element of degree 4

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

1. b_1_0 → 0, an element of degree 1
2. b_2_2 → 0, an element of degree 2
3. b_2_0 → c_1_2² + c_1_1·c_1_2 + c_1_1², an element of degree 2
4. b_3_4 → 0, an element of degree 3
5. b_3_2 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
6. b_3_1 → c_1_2³ + c_1_1²·c_1_2 + c_1_1³, an element of degree 3
7. b_3_0 → 0, an element of degree 3
8. c_4_6 → 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

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

1. b_1_0 → 0, an element of degree 1
2. b_2_2 → c_1_2² + c_1_1·c_1_2, an element of degree 2
3. b_2_0 → c_1_1², an element of degree 2
4. b_3_4 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
5. b_3_2 → 0, an element of degree 3
6. b_3_1 → c_1_1³, an element of degree 3
7. b_3_0 → 0, an element of degree 3
8. c_4_6 → 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

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

1. b_1_0 → 0, an element of degree 1
2. b_2_2 → c_1_2² + c_1_1·c_1_2, an element of degree 2
3. b_2_0 → c_1_1², an element of degree 2
4. b_3_4 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
5. b_3_2 → 0, an element of degree 3
6. b_3_1 → c_1_1³, an element of degree 3
7. b_3_0 → 0, an element of degree 3
8. c_4_6 → 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

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

1. b_1_0 → 0, an element of degree 1
2. b_2_2 → c_1_2² + c_1_1·c_1_2 + c_1_1², an element of degree 2
3. b_2_0 → 0, an element of degree 2
4. b_3_4 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
5. b_3_2 → 0, an element of degree 3
6. b_3_1 → 0, an element of degree 3
7. b_3_0 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
8. c_4_6 → 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

About the group

Ring generators

Ring relations

Completion information

Restriction maps