David J. Green

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Cohomology of group number 1136 of order 128

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Ring relations

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General information on the group

• The group has 4 minimal generators and exponent 4.
• It is non-abelian.
• It has p-Rank 5.
• Its center has rank 3.
• It has 3 conjugacy classes of maximal elementary abelian subgroups, which are of rank 4, 4 and 5, respectively.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 5 and depth 4.
• The depth exceeds the Duflot bound, which is 3.
• The Poincaré series is

  t3  −  t2  −  1

  (t  +  1) · (t  −  1)5 · (t2  +  1)

• The a-invariants are -∞,-∞,-∞,-∞,-5,-5. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 10 minimal generators of maximal degree 4:

1. b_1_0, an element of degree 1
2. b_1_1, an element of degree 1
3. b_1_2, an element of degree 1
4. b_1_3, an element of degree 1
5. b_2_7, an element of degree 2
6. b_2_8, an element of degree 2
7. c_2_9, a Duflot regular element of degree 2
8. c_2_10, a Duflot regular element of degree 2
9. b_3_22, an element of degree 3
10. c_4_41, a Duflot regular element of degree 4

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Ring relations

There are 14 minimal relations of maximal degree 6:

1. b_1_0·b_1_2
2. b_1_1·b_1_2
3. b_1_1² + b_1_0·b_1_3 + b_1_0·b_1_1
4. b_1_0·b_1_3²
5. b_2_7·b_1_2
6. b_2_8·b_1_0 + b_2_7·b_1_1 + b_2_7·b_1_0
7. b_2_8·b_1_1 + b_2_7·b_1_3
8. b_2_7·b_1_0·b_1_1 + b_2_7² + c_2_10·b_1_0² + c_2_9·b_1_0·b_1_3 + c_2_9·b_1_0·b_1_1
9. b_2_7·b_1_0·b_1_3 + b_2_7·b_2_8 + c_2_10·b_1_0·b_1_1 + c_2_10·b_1_0²
      + c_2_9·b_1_1·b_1_3
10. b_1_2·b_3_22 + b_2_8² + b_2_7·b_1_1·b_1_3 + b_2_7·b_1_0·b_1_3 + c_2_10·b_1_0·b_1_3
       + c_2_10·b_1_0·b_1_1 + c_2_10·b_1_0² + c_2_9·b_1_3²
11. b_1_0·b_3_22
12. b_1_1·b_3_22 + b_2_7·b_1_3²
13. b_2_7·b_3_22 + c_2_9·b_1_1·b_1_3²
14. b_3_22² + b_2_8²·b_1_3² + c_4_41·b_1_2²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 7.
• 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_2_9, a Duflot regular element of degree 2
  2. c_2_10, a Duflot regular element of degree 2
  3. c_4_41, a Duflot regular element of degree 4
  4. b_1_3² + b_1_2·b_1_3 + b_1_2² + b_1_0², an element of degree 2
  5. b_1_3², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 5, 7].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 3

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. b_2_7 → 0, an element of degree 2
6. b_2_8 → 0, an element of degree 2
7. c_2_9 → c_1_1², an element of degree 2
8. c_2_10 → c_1_2², an element of degree 2
9. b_3_22 → 0, an element of degree 3
10. c_4_41 → c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

1. b_1_0 → c_1_3, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. b_2_7 → c_1_2·c_1_3, an element of degree 2
6. b_2_8 → c_1_2·c_1_3, an element of degree 2
7. c_2_9 → c_1_1·c_1_3 + c_1_1², an element of degree 2
8. c_2_10 → c_1_2², an element of degree 2
9. b_3_22 → 0, an element of degree 3
10. c_4_41 → c_1_0²·c_1_3² + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

1. b_1_0 → c_1_3, an element of degree 1
2. b_1_1 → c_1_3, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. b_2_7 → c_1_2·c_1_3 + c_1_1·c_1_3, an element of degree 2
6. b_2_8 → 0, an element of degree 2
7. c_2_9 → c_1_1·c_1_3 + c_1_1², an element of degree 2
8. c_2_10 → c_1_2·c_1_3 + c_1_2², an element of degree 2
9. b_3_22 → 0, an element of degree 3
10. c_4_41 → c_1_3⁴ + c_1_0²·c_1_3² + c_1_0⁴, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → c_1_3, an element of degree 1
4. b_1_3 → c_1_4, an element of degree 1
5. b_2_7 → 0, an element of degree 2
6. b_2_8 → c_1_1·c_1_4 + c_1_0·c_1_3, an element of degree 2
7. c_2_9 → c_1_1·c_1_3 + c_1_1², an element of degree 2
8. c_2_10 → c_1_2·c_1_3 + c_1_2², an element of degree 2
9. b_3_22 → c_1_1·c_1_4² + c_1_0²·c_1_3, an element of degree 3
10. c_4_41 → c_1_0²·c_1_4² + c_1_0⁴, an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128