David J. Green

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Cohomology of group number 734 of order 128

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General information on the group

• The group has 3 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 4.
• Its center has rank 2.
• It has 4 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 4, 4 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 2.
• The Poincaré series is

  1

  (t  +  1) · (t  −  1)4

• The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 11 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_2_3, an element of degree 2
5. b_2_4, an element of degree 2
6. b_2_6, an element of degree 2
7. c_2_5, a Duflot regular element of degree 2
8. b_3_11, an element of degree 3
9. b_3_12, an element of degree 3
10. b_4_17, an element of degree 4
11. c_4_21, a Duflot regular element of degree 4

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Ring relations

There are 27 minimal relations of maximal degree 8:

1. b_1_0·b_1_1
2. b_1_0·b_1_2
3. b_1_1·b_1_2
4. b_2_3·b_1_1
5. b_2_4·b_1_2
6. b_2_4·b_1_0 + b_2_3·b_1_0
7. b_2_6·b_1_0
8. b_2_3·b_1_2² + b_2_3²
9. b_2_3·b_2_4
10. b_2_4² + c_2_5·b_1_1²
11. b_1_2·b_3_11 + b_2_3·b_2_6
12. b_1_0·b_3_11
13. b_1_0·b_3_12
14. b_1_1·b_3_12 + b_1_1·b_3_11 + b_2_4·b_2_6
15. b_2_3·b_3_11 + b_2_3·b_2_6·b_1_2
16. b_2_4·b_3_12 + b_2_4·b_3_11 + b_2_6·c_2_5·b_1_1
17. b_4_17·b_1_2 + b_2_3·b_3_12 + b_2_3·b_2_6·b_1_2 + b_2_3·c_2_5·b_1_2
18. b_4_17·b_1_0 + b_2_3·c_2_5·b_1_0
19. b_4_17·b_1_1 + b_2_4·b_3_11 + b_2_4·b_2_6·b_1_1 + c_2_5·b_1_1³
20. b_3_12² + b_3_11² + b_2_6·b_1_2·b_3_12 + b_2_3·b_2_6² + c_4_21·b_1_2²
       + b_2_6·c_2_5·b_1_2² + b_2_6²·c_2_5
21. b_3_11² + b_1_1³·b_3_11 + b_2_6·b_1_1·b_3_11 + b_2_3·b_2_6² + c_4_21·b_1_1²
       + b_2_6·c_2_5·b_1_1²
22. b_2_3·b_1_2·b_3_12 + b_2_3·b_4_17 + b_2_3²·b_2_6 + b_2_3²·c_2_5
23. b_2_4·b_4_17 + c_2_5·b_1_1·b_3_11 + b_2_6·c_2_5·b_1_1² + b_2_4·c_2_5·b_1_1²
24. b_3_11·b_3_12 + b_3_11² + b_2_6·b_4_17 + b_2_4·b_2_6² + b_2_6·c_2_5·b_1_1²
       + b_2_3·b_2_6·c_2_5
25. b_4_17·b_3_12 + b_2_4·b_1_1²·b_3_11 + c_2_5·b_1_1²·b_3_11 + b_2_6·c_2_5·b_3_11
       + b_2_6²·c_2_5·b_1_1 + b_2_4·c_4_21·b_1_1 + b_2_3·c_4_21·b_1_2 + b_2_3·c_2_5·b_3_12
       + b_2_3·b_2_6·c_2_5·b_1_2
26. b_4_17·b_3_11 + b_2_4·b_1_1²·b_3_11 + b_2_3·b_2_6·b_3_12 + b_2_3·b_2_6²·b_1_2
       + c_2_5·b_1_1²·b_3_11 + b_2_4·c_4_21·b_1_1 + b_2_4·b_2_6·c_2_5·b_1_1
       + b_2_3·b_2_6·c_2_5·b_1_2
27. b_4_17² + b_2_3·b_2_6·b_4_17 + c_2_5·b_1_1³·b_3_11 + b_2_6·c_2_5·b_1_1·b_3_11
       + b_2_6²·c_2_5·b_1_1² + b_2_3·b_2_6²·c_2_5 + b_2_3²·c_4_21 + c_2_5·c_4_21·b_1_1²
       + c_2_5²·b_1_1⁴ + b_2_6·c_2_5²·b_1_1² + b_2_3²·c_2_5²

About the group

Ring generators

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 8.
• 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_2_5, a Duflot regular element of degree 2
  2. c_4_21, a Duflot regular element of degree 4
  3. b_1_2² + b_1_1² + b_1_0² + b_2_6, an element of degree 2
  4. b_1_2² + b_1_1², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 4, 6].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

About the group

Ring generators

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

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_2_3 → 0, an element of degree 2
5. b_2_4 → 0, an element of degree 2
6. b_2_6 → 0, an element of degree 2
7. c_2_5 → c_1_0², an element of degree 2
8. b_3_11 → 0, an element of degree 3
9. b_3_12 → 0, an element of degree 3
10. b_4_17 → 0, an element of degree 4
11. c_4_21 → c_1_1⁴, an element of degree 4

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

1. b_1_0 → c_1_2, 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_2_3 → 0, an element of degree 2
5. b_2_4 → 0, an element of degree 2
6. b_2_6 → 0, an element of degree 2
7. c_2_5 → c_1_0·c_1_2 + c_1_0², an element of degree 2
8. b_3_11 → 0, an element of degree 3
9. b_3_12 → 0, an element of degree 3
10. b_4_17 → 0, an element of degree 4
11. c_4_21 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4

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

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_2, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_2_3 → 0, an element of degree 2
5. b_2_4 → c_1_0·c_1_2, an element of degree 2
6. b_2_6 → c_1_3² + c_1_2·c_1_3, an element of degree 2
7. c_2_5 → c_1_0², an element of degree 2
8. b_3_11 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
9. b_3_12 → c_1_1·c_1_2² + c_1_1²·c_1_2 + c_1_0·c_1_3² + c_1_0·c_1_2·c_1_3, an element of degree 3
10. b_4_17 → 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_2²
       + c_1_0·c_1_1²·c_1_2 + c_1_0²·c_1_2², an element of degree 4
11. c_4_21 → c_1_1·c_1_2·c_1_3² + c_1_1·c_1_2²·c_1_3 + c_1_1·c_1_2³ + c_1_1²·c_1_3²
       + c_1_1²·c_1_2·c_1_3 + c_1_1⁴ + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3, an element of degree 4

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

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_2, an element of degree 1
4. b_2_3 → 0, an element of degree 2
5. b_2_4 → 0, an element of degree 2
6. b_2_6 → c_1_3² + c_1_2·c_1_3, an element of degree 2
7. c_2_5 → c_1_0·c_1_2 + c_1_0², an element of degree 2
8. b_3_11 → 0, an element of degree 3
9. b_3_12 → c_1_1·c_1_2² + c_1_1²·c_1_2 + c_1_0·c_1_3² + c_1_0·c_1_2·c_1_3, an element of degree 3
10. b_4_17 → 0, an element of degree 4
11. c_4_21 → c_1_1·c_1_2·c_1_3² + c_1_1·c_1_2²·c_1_3 + c_1_1²·c_1_3² + c_1_1²·c_1_2·c_1_3
       + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3
       + c_1_0²·c_1_3² + c_1_0²·c_1_2·c_1_3, an element of degree 4

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

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_2_3 → c_1_3², an element of degree 2
5. b_2_4 → 0, an element of degree 2
6. b_2_6 → c_1_2·c_1_3 + c_1_2², an element of degree 2
7. c_2_5 → c_1_0·c_1_3 + c_1_0², an element of degree 2
8. b_3_11 → c_1_2·c_1_3² + c_1_2²·c_1_3, an element of degree 3
9. b_3_12 → c_1_2·c_1_3² + c_1_2²·c_1_3 + c_1_1·c_1_3² + c_1_1²·c_1_3 + c_1_0·c_1_2·c_1_3
      + c_1_0·c_1_2², an element of degree 3
10. b_4_17 → c_1_1·c_1_3³ + c_1_1²·c_1_3² + c_1_0·c_1_3³ + c_1_0·c_1_2·c_1_3²
       + c_1_0·c_1_2²·c_1_3 + c_1_0²·c_1_3², an element of degree 4
11. c_4_21 → c_1_1·c_1_2·c_1_3² + c_1_1·c_1_2²·c_1_3 + c_1_1²·c_1_3² + c_1_1²·c_1_2·c_1_3
       + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0·c_1_2·c_1_3² + c_1_0·c_1_2²·c_1_3
       + c_1_0²·c_1_2·c_1_3 + c_1_0²·c_1_2², an element of degree 4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128