David J. Green

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Cohomology of group number 1736 of order 128

About the group

Ring generators

Ring relations

Completion information

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General information on the group

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

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  t5  +  t4  −  t3  +  t2  +  1

  (t  −  1)4 · (t2  +  1) · (t4  +  1)

• The a-invariants are -∞,-∞,-6,-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 8:

1. a_1_1, a nilpotent element of degree 1
2. b_1_0, an element of degree 1
3. b_1_2, an element of degree 1
4. b_1_3, an element of degree 1
5. a_2_8, a nilpotent element of degree 2
6. c_2_9, a Duflot regular element of degree 2
7. a_5_46, a nilpotent element of degree 5
8. b_5_45, an element of degree 5
9. b_5_47, an element of degree 5
10. a_6_69, a nilpotent element of degree 6
11. c_8_135, a Duflot regular element of degree 8

About the group

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Ring relations

There are 27 minimal relations of maximal degree 12:

1. a_1_1·b_1_0
2. b_1_0·b_1_2
3. b_1_0·b_1_3² + a_1_1³
4. a_2_8·b_1_0
5. a_1_1²·b_1_2² + a_2_8·a_1_1·b_1_2 + a_2_8² + c_2_9·a_1_1²
6. a_1_1³·b_1_3²
7. b_1_0·a_5_46
8. b_1_2·b_5_45 + a_2_8·a_1_1²·b_1_3²
9. a_1_1·b_5_45
10. b_1_2·a_5_46 + a_1_1·b_5_47 + a_2_8·b_1_3⁴
11. b_1_0·b_5_47
12. b_1_3²·b_5_45 + a_1_1²·a_5_46
13. a_2_8·b_5_45
14. a_6_69·b_1_2 + a_2_8·b_5_47 + a_2_8²·b_1_3³ + a_2_8²·a_1_1·b_1_3²
       + c_2_9·a_1_1·b_1_3⁴ + c_2_9·a_1_1²·b_1_3³
15. a_1_1²·b_1_2·b_1_3⁴ + a_6_69·a_1_1 + a_2_8·a_5_46 + a_2_8·a_1_1·b_1_3⁴
       + a_2_8·a_1_1²·b_1_3³
16. a_6_69·b_1_0
17. a_1_1²·b_1_2·b_5_47 + a_2_8·a_1_1·b_5_47 + a_2_8·a_6_69 + c_2_9·a_1_1·a_5_46
18. a_5_46·b_5_45
19. b_5_45·b_5_47 + a_2_8·a_1_1·b_1_3²·a_5_46 + a_2_8·a_1_1²·b_1_3⁶
20. b_5_47² + b_1_2·b_1_3⁴·b_5_47 + a_1_1·b_1_2·b_1_3³·b_5_47 + a_2_8·b_1_2·b_1_3⁷
       + a_2_8²·a_1_1·b_1_3⁵ + c_8_135·b_1_2² + c_2_9·b_1_3⁸
21. a_5_46·b_5_47 + a_1_1·b_1_3⁴·b_5_47 + a_6_69·b_1_3⁴ + a_1_1²·b_1_3³·b_5_47
       + a_6_69·a_1_1·b_1_3³ + a_2_8·b_1_3³·a_5_46 + a_2_8·a_1_1·b_1_3⁷
       + c_8_135·a_1_1·b_1_2
22. b_5_45² + c_8_135·b_1_0²
23. a_5_46² + a_1_1·b_1_3⁴·a_5_46 + a_1_1²·b_1_3⁸ + a_1_1²·b_1_3³·a_5_46
       + c_8_135·a_1_1²
24. a_6_69·b_5_45
25. a_6_69·b_5_47 + a_2_8·b_1_3⁴·b_5_47 + a_2_8·a_1_1·b_1_3³·b_5_47
       + a_2_8·a_6_69·b_1_3³ + a_2_8²·b_1_3⁷ + a_2_8²·b_1_3²·a_5_46
       + c_2_9·b_1_3⁴·a_5_46 + a_2_8·c_8_135·b_1_2 + c_2_9·a_1_1·b_1_3³·a_5_46
       + c_2_9·a_1_1²·b_1_3²·a_5_46
26. a_1_1²·b_1_3⁴·b_5_47 + a_6_69·a_5_46 + a_2_8·a_1_1²·b_1_3⁷ + a_2_8·c_8_135·a_1_1
27. a_6_69² + a_2_8·a_6_69·b_1_3⁴ + a_2_8²·b_1_3³·a_5_46 + c_2_9·a_1_1²·b_1_3⁸
       + a_2_8²·c_8_135

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Data used for Benson′s test

• Benson′s completion test succeeded in degree 12.
• 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_9, a Duflot regular element of degree 2
  2. c_8_135, a Duflot regular element of degree 8
  3. b_1_3² + b_1_2·b_1_3 + b_1_2² + b_1_0², an element of degree 2
  4. b_1_3², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 8, 10].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

About the group

Ring generators

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Completion information

Restriction maps

Back to groups of order 128

Restriction maps

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

1. a_1_1 → 0, an element of degree 1
2. b_1_0 → 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. a_2_8 → 0, an element of degree 2
6. c_2_9 → c_1_0², an element of degree 2
7. a_5_46 → 0, an element of degree 5
8. b_5_45 → 0, an element of degree 5
9. b_5_47 → 0, an element of degree 5
10. a_6_69 → 0, an element of degree 6
11. c_8_135 → c_1_1⁸, an element of degree 8

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

1. a_1_1 → 0, an element of degree 1
2. b_1_0 → c_1_2, 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. a_2_8 → 0, an element of degree 2
6. c_2_9 → c_1_0·c_1_2 + c_1_0², an element of degree 2
7. a_5_46 → 0, an element of degree 5
8. b_5_45 → c_1_1²·c_1_2³ + c_1_1⁴·c_1_2, an element of degree 5
9. b_5_47 → 0, an element of degree 5
10. a_6_69 → 0, an element of degree 6
11. c_8_135 → c_1_1⁴·c_1_2⁴ + c_1_1⁸, an element of degree 8

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

1. a_1_1 → 0, an element of degree 1
2. b_1_0 → 0, an element of degree 1
3. b_1_2 → c_1_2, an element of degree 1
4. b_1_3 → c_1_3, an element of degree 1
5. a_2_8 → 0, an element of degree 2
6. c_2_9 → c_1_0·c_1_2 + c_1_0², an element of degree 2
7. a_5_46 → 0, an element of degree 5
8. b_5_45 → 0, an element of degree 5
9. b_5_47 → c_1_1⁴·c_1_2 + c_1_0·c_1_3⁴, an element of degree 5
10. a_6_69 → 0, an element of degree 6
11. c_8_135 → c_1_1⁴·c_1_3⁴ + c_1_1⁸, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128