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Singular

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Mod-2-Cohomology of group number 1493 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 4.
• 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 of rank 3, 3, 3, 3 and 4, respectively.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 3.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  1  −  t  +  3·t2  −  t4  +  t5

  (1  +  t) · ( − 1  +  t)4 · (1  +  t2) · (1  +  t  +  t2)

• The a-invariants are -∞,-∞,-∞,-6,-4. 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, -4, -4].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 9 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_1, an element of degree 2
4. b_2_0, an element of degree 2
5. b_3_6, an element of degree 3
6. b_3_5, an element of degree 3
7. b_3_2, an element of degree 3
8. b_3_0, an element of degree 3
9. c_4_10, a Duflot element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 17 minimal relations of maximal degree 6:

1. b_1_0·b_3_0 + b_2_1² + b_2_0·b_2_2
2. b_1_0·b_3_2
3. b_1_0·b_3_5
4. b_1_0·b_3_6
5. b_2_0·b_3_6 + b_2_0·b_3_5
6. b_2_1·b_3_2 + b_2_0·b_3_5
7. b_2_1·b_3_5 + b_2_0·b_3_5
8. b_2_1·b_3_6 + b_2_0·b_3_5
9. b_2_2·b_3_2 + b_2_0·b_3_5
10. b_2_2·b_3_5 + b_2_0·b_3_5
11. b_3_0² + b_2_1²·b_2_2 + b_2_1³ + b_2_0·b_2_1·b_2_2 + b_2_0·b_2_1² + c_4_10·b_1_0²
12. b_3_0·b_3_2
13. b_3_0·b_3_5
14. b_3_0·b_3_6
15. b_3_2·b_3_6 + b_3_2·b_3_5
16. b_3_5² + b_3_2·b_3_5
17. b_3_5·b_3_6 + b_3_2·b_3_5

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 7 using the Hilbert-Poincaré criterion.
• However, the last relation was already found in degree 6 and the last generator in degree 4.
• The following is a filter regular homogeneous system of parameters:
  1. c_4_10, an element of degree 4
  2. b_1_0⁴ + b_2_2² + b_2_0·b_2_2 + b_2_0², an element of degree 4
  3. b_3_6² + b_3_2·b_3_5 + b_3_2² + b_2_2²·b_1_0² + b_2_0·b_2_2·b_1_0²
        + b_2_0·b_2_2² + b_2_0²·b_1_0² + b_2_0²·b_2_2, an element of degree 6
  4. b_1_0, an element of degree 1
• A Duflot regular sequence is given by c_4_10.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 8, 11].
• Modifying the above filter regular HSOP, we obtained the following parameters:
1. c_4_10, an element of degree 4
2. b_2_2 + b_2_1 + b_2_0, an element of degree 2
3. b_3_6 + b_3_5 + b_3_2 + b_3_0, an element of degree 3
4. b_1_0, an element of degree 1

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(64,138); GF(2))

1. b_1_0 → b_1_0
2. b_2_2 → b_1_1² + b_2_4
3. b_2_1 → b_1_1·b_1_2 + b_2_5
4. b_2_0 → b_1_2² + b_2_6
5. b_3_6 → b_2_4·b_1_1
6. b_3_5 → b_2_4·b_1_2
7. b_3_2 → b_2_6·b_1_2
8. b_3_0 → b_3_11 + b_1_1·b_1_2² + b_1_1²·b_1_2
9. c_4_10 → c_4_18

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_1 → 0, an element of degree 2
4. b_2_0 → 0, an element of degree 2
5. b_3_6 → 0, an element of degree 3
6. b_3_5 → 0, an element of degree 3
7. b_3_2 → 0, an element of degree 3
8. b_3_0 → 0, an element of degree 3
9. c_4_10 → 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_1², an element of degree 2
3. b_2_1 → c_1_1·c_1_2, an element of degree 2
4. b_2_0 → c_1_2², an element of degree 2
5. b_3_6 → 0, an element of degree 3
6. b_3_5 → 0, an element of degree 3
7. b_3_2 → 0, an element of degree 3
8. b_3_0 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
9. c_4_10 → 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_1 → 0, an element of degree 2
4. b_2_0 → 0, an element of degree 2
5. b_3_6 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
6. b_3_5 → 0, an element of degree 3
7. b_3_2 → 0, an element of degree 3
8. b_3_0 → 0, an element of degree 3
9. c_4_10 → 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 → 0, an element of degree 2
3. b_2_1 → 0, an element of degree 2
4. b_2_0 → c_1_2² + c_1_1·c_1_2 + c_1_1², an element of degree 2
5. b_3_6 → 0, an element of degree 3
6. b_3_5 → 0, an element of degree 3
7. b_3_2 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
8. b_3_0 → 0, an element of degree 3
9. c_4_10 → 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_1 → c_1_2² + c_1_1·c_1_2 + c_1_1², an element of degree 2
4. b_2_0 → c_1_2² + c_1_1·c_1_2 + c_1_1², an element of degree 2
5. b_3_6 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
6. b_3_5 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
7. b_3_2 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
8. b_3_0 → 0, an element of degree 3
9. c_4_10 → 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 4 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_1 → c_1_2·c_1_3 + c_1_0·c_1_1, an element of degree 2
4. b_2_0 → c_1_3² + c_1_1·c_1_3, an element of degree 2
5. b_3_6 → 0, an element of degree 3
6. b_3_5 → 0, an element of degree 3
7. b_3_2 → 0, an element of degree 3
8. b_3_0 → c_1_2·c_1_3² + c_1_2²·c_1_3 + c_1_1·c_1_2·c_1_3 + c_1_0²·c_1_1, an element of degree 3
9. c_4_10 → c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3 + c_1_0·c_1_1·c_1_2·c_1_3
      + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3 + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_3
      + c_1_0²·c_1_1·c_1_2 + c_1_0³·c_1_1 + c_1_0⁴, an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps