David J. Green

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Cohomology of group number 849 of order 128

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Ring generators

Ring relations

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General information on the group

• The group has 3 minimal generators and exponent 16.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 1.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 2.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  ( − 1) · (t5  +  t2  +  1)

  (t  −  1)3 · (t2  +  1) · (t4  +  1)

• The a-invariants are -∞,-∞,-3,-3. 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 8:

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. a_2_4, a nilpotent element of degree 2
5. b_2_5, an element of degree 2
6. b_3_9, an element of degree 3
7. a_5_12, a nilpotent element of degree 5
8. b_5_17, an element of degree 5
9. a_6_15, a nilpotent element of degree 6
10. c_8_37, a Duflot regular element of degree 8

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Ring relations

There are 27 minimal relations of maximal degree 12:

1. a_1_0²
2. a_1_0·b_1_1
3. a_2_4·a_1_0
4. b_2_5·b_1_1 + a_1_0·b_1_2² + a_2_4·b_1_1
5. a_2_4²
6. a_1_0·b_3_9 + a_2_4·b_2_5
7. b_1_1·b_3_9 + a_2_4·b_1_2²
8. a_1_0·b_1_2⁴ + b_2_5·a_1_0·b_1_2²
9. b_2_5²·a_1_0 + a_2_4·b_3_9
10. b_3_9² + b_2_5³ + a_2_4·b_2_5·b_1_2² + a_2_4·b_2_5²
11. a_1_0·a_5_12
12. b_1_1·a_5_12 + a_2_4·b_1_2⁴ + a_2_4·b_2_5·b_1_2²
13. a_1_0·b_5_17 + a_2_4·b_1_2·b_3_9 + a_2_4·b_2_5·b_1_2²
14. a_2_4·a_5_12
15. b_1_2⁴·b_3_9 + b_2_5·b_5_17 + b_2_5³·b_1_2 + b_1_2²·a_5_12 + a_2_4·b_5_17
       + a_2_4·b_1_2²·b_3_9 + a_2_4·b_2_5²·b_1_2
16. a_6_15·a_1_0
17. b_1_2⁴·b_3_9 + b_2_5·b_5_17 + b_2_5³·b_1_2 + b_1_2²·a_5_12 + a_6_15·b_1_1
       + a_2_4·b_1_2⁵ + a_2_4·b_1_1²·b_1_2³ + a_2_4·b_2_5·b_1_2³
18. b_3_9·a_5_12 + b_2_5·b_1_2·a_5_12 + b_2_5·a_6_15 + a_2_4·b_2_5·b_1_2·b_3_9
19. b_3_9·b_5_17 + b_2_5²·b_1_2·b_3_9 + b_2_5²·b_1_2⁴ + b_1_2³·a_5_12 + a_6_15·b_1_2²
       + a_2_4·b_1_2³·b_3_9 + a_2_4·b_1_1·b_1_2⁵
20. a_2_4·a_6_15
21. a_6_15·b_3_9 + b_2_5·b_1_2²·a_5_12 + b_2_5·a_6_15·b_1_2 + b_2_5²·a_5_12
       + a_2_4·b_2_5·b_1_2²·b_3_9 + a_2_4·b_2_5³·b_1_2
22. a_5_12²
23. a_5_12·b_5_17 + b_1_2⁵·a_5_12 + a_6_15·b_1_2⁴ + b_2_5²·b_1_2·a_5_12
       + a_2_4·b_1_1·b_1_2⁷ + a_2_4·b_2_5·b_1_2³·b_3_9
24. b_5_17² + b_1_1·b_1_2⁴·b_5_17 + b_1_1²·b_1_2³·b_5_17 + b_1_1³·b_1_2²·b_5_17
       + b_1_1⁴·b_1_2·b_5_17 + b_2_5·b_1_2⁸ + b_2_5⁴·b_1_2² + a_2_4·b_1_2⁸
       + a_2_4·b_1_1·b_1_2⁷ + a_2_4·b_1_1³·b_1_2⁵ + a_2_4·b_1_1⁴·b_1_2⁴
       + a_2_4·b_1_1⁵·b_1_2³ + a_2_4·b_2_5·b_1_2³·b_3_9 + c_8_37·b_1_1²
25. a_6_15·a_5_12
26. b_1_2⁶·a_5_12 + a_6_15·b_5_17 + a_6_15·b_1_2⁵ + b_2_5·b_1_2⁴·a_5_12
       + b_2_5²·a_6_15·b_1_2 + a_2_4·b_1_2⁴·b_5_17 + a_2_4·b_1_1·b_1_2⁸
       + a_2_4·b_1_1²·b_1_2²·b_5_17 + a_2_4·b_1_1³·b_1_2·b_5_17 + a_2_4·c_8_37·b_1_1
27. a_6_15²

About the group

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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_8_37, a Duflot regular element of degree 8
  2. b_1_2⁴ + b_1_1²·b_1_2² + b_1_1⁴ + b_2_5·b_1_2² + b_2_5², an element of degree 4
  3. b_1_2², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 11].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group

Ring generators

Ring relations

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 1

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. a_2_4 → 0, an element of degree 2
5. b_2_5 → 0, an element of degree 2
6. b_3_9 → 0, an element of degree 3
7. a_5_12 → 0, an element of degree 5
8. b_5_17 → 0, an element of degree 5
9. a_6_15 → 0, an element of degree 6
10. c_8_37 → c_1_0⁸, an element of degree 8

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

1. a_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_1, an element of degree 1
3. b_1_2 → c_1_2, an element of degree 1
4. a_2_4 → 0, an element of degree 2
5. b_2_5 → 0, an element of degree 2
6. b_3_9 → 0, an element of degree 3
7. a_5_12 → 0, an element of degree 5
8. b_5_17 → c_1_0²·c_1_1³ + c_1_0⁴·c_1_1, an element of degree 5
9. a_6_15 → 0, an element of degree 6
10. c_8_37 → c_1_0²·c_1_1²·c_1_2⁴ + c_1_0²·c_1_1³·c_1_2³ + c_1_0²·c_1_1⁴·c_1_2²
       + c_1_0²·c_1_1⁵·c_1_2 + c_1_0⁴·c_1_2⁴ + c_1_0⁴·c_1_1·c_1_2³
       + c_1_0⁴·c_1_1²·c_1_2² + c_1_0⁴·c_1_1³·c_1_2 + c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8

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

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_1, an element of degree 1
4. a_2_4 → 0, an element of degree 2
5. b_2_5 → c_1_2², an element of degree 2
6. b_3_9 → c_1_2³, an element of degree 3
7. a_5_12 → 0, an element of degree 5
8. b_5_17 → c_1_1·c_1_2⁴ + c_1_1⁴·c_1_2, an element of degree 5
9. a_6_15 → 0, an element of degree 6
10. c_8_37 → c_1_2⁸ + c_1_1·c_1_2⁷ + c_1_1²·c_1_2⁶ + c_1_1⁵·c_1_2³ + c_1_1⁶·c_1_2²
       + c_1_1⁷·c_1_2 + c_1_0²·c_1_1²·c_1_2⁴ + c_1_0²·c_1_1⁴·c_1_2²
       + c_1_0⁴·c_1_2⁴ + c_1_0⁴·c_1_1²·c_1_2² + c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128