David J. Green

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Cohomology of group number 170 of order 128

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General information on the group

• The group has 3 minimal generators and exponent 4.
• It is non-abelian.
• It has p-Rank 5.
• Its center has rank 4.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 5.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 5 and depth 4.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  ( − 2) · (t2  +  1/2·t  +  1/2)

  (t  +  1)3 · (t  −  1)5

• 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 12 minimal generators of maximal degree 3:

1. a_1_0, a nilpotent 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_2, an element of degree 2
5. b_2_3, an element of degree 2
6. b_2_4, an element of degree 2
7. b_2_5, an element of degree 2
8. c_2_6, a Duflot regular element of degree 2
9. c_2_7, a Duflot regular element of degree 2
10. c_2_8, a Duflot regular element of degree 2
11. c_2_9, a Duflot regular element of degree 2
12. b_3_19, an element of degree 3

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Ring relations

There are 27 minimal relations of maximal degree 6:

1. a_1_0²
2. a_1_0·b_1_1
3. a_1_0·b_1_2
4. b_1_1·b_1_2
5. b_2_2·a_1_0
6. b_2_3·a_1_0
7. b_2_3·b_1_1 + b_2_2·b_1_2
8. b_2_2·b_1_2 + b_2_4·a_1_0
9. b_2_4·b_1_1
10. b_2_5·b_1_2 + b_2_4·b_1_2
11. b_2_5·a_1_0
12. b_2_2² + c_2_6·b_1_1²
13. b_2_3² + c_2_6·b_1_2²
14. b_2_2·b_2_3
15. b_2_2·b_2_4
16. b_2_4² + c_2_7·b_1_2²
17. b_2_3·b_2_5 + b_2_3·b_2_4
18. b_2_5² + c_2_8·b_1_1² + c_2_7·b_1_2²
19. b_2_4·b_2_5 + c_2_7·b_1_2²
20. b_1_2·b_3_19 + b_2_3·b_2_4 + c_2_7·b_1_2²
21. a_1_0·b_3_19
22. b_1_1·b_3_19 + b_2_2·b_2_5
23. b_2_3·b_3_19 + b_2_4·c_2_6·b_1_2 + b_2_3·c_2_7·b_1_2
24. b_2_2·b_3_19 + b_2_5·c_2_6·b_1_1 + b_2_4·c_2_7·a_1_0
25. b_2_5·b_3_19 + b_2_4·c_2_7·b_1_2 + b_2_3·c_2_7·b_1_2 + b_2_2·c_2_8·b_1_1
26. b_2_4·b_3_19 + b_2_4·c_2_7·b_1_2 + b_2_3·c_2_7·b_1_2
27. b_3_19² + c_2_7²·b_1_2² + c_2_6·c_2_8·b_1_1² + c_2_6·c_2_7·b_1_2²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 6.
• 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_6, a Duflot regular element of degree 2
  2. c_2_7, a Duflot regular element of degree 2
  3. c_2_8, a Duflot regular element of degree 2
  4. c_2_9, a Duflot regular element of degree 2
  5. b_1_2² + b_1_1², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 3, 5].
• 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 4

1. a_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_2 → 0, an element of degree 2
5. b_2_3 → 0, an element of degree 2
6. b_2_4 → 0, an element of degree 2
7. b_2_5 → 0, an element of degree 2
8. c_2_6 → c_1_3², an element of degree 2
9. c_2_7 → c_1_0², an element of degree 2
10. c_2_8 → c_1_1², an element of degree 2
11. c_2_9 → c_1_2², an element of degree 2
12. b_3_19 → 0, an element of degree 3

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

1. a_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_4, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_2_2 → c_1_3·c_1_4, an element of degree 2
5. b_2_3 → 0, an element of degree 2
6. b_2_4 → 0, an element of degree 2
7. b_2_5 → c_1_1·c_1_4, an element of degree 2
8. c_2_6 → c_1_3², an element of degree 2
9. c_2_7 → c_1_0·c_1_4 + c_1_0², an element of degree 2
10. c_2_8 → c_1_1², an element of degree 2
11. c_2_9 → c_1_2·c_1_4 + c_1_2², an element of degree 2
12. b_3_19 → c_1_1·c_1_3·c_1_4, an element of degree 3

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

1. a_1_0 → 0, an element of degree 1
2. b_1_1 → 0, an element of degree 1
3. b_1_2 → c_1_4, an element of degree 1
4. b_2_2 → 0, an element of degree 2
5. b_2_3 → c_1_3·c_1_4, an element of degree 2
6. b_2_4 → c_1_0·c_1_4, an element of degree 2
7. b_2_5 → c_1_0·c_1_4, an element of degree 2
8. c_2_6 → c_1_3², an element of degree 2
9. c_2_7 → c_1_0², an element of degree 2
10. c_2_8 → c_1_1·c_1_4 + c_1_1², an element of degree 2
11. c_2_9 → c_1_2·c_1_4 + c_1_2², an element of degree 2
12. b_3_19 → c_1_0·c_1_3·c_1_4 + c_1_0²·c_1_4, an element of degree 3

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128