David J. Green

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Cohomology of group number 1688 of order 128

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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 3.
• Its center has rank 1.
• It has 3 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) · (t6  −  t5  +  t2  +  t  +  1)

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

• The a-invariants are -∞,-∞,-4,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 9 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. b_1_3, an element of degree 1
5. b_2_8, an element of degree 2
6. a_6_27, a nilpotent element of degree 6
7. a_6_13, a nilpotent element of degree 6
8. a_7_25, a nilpotent element of degree 7
9. c_8_43, a Duflot regular element of degree 8

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Ring relations

There are 18 minimal relations of maximal degree 14:

1. a_1_0²
2. b_1_1·b_1_2 + a_1_0·b_1_1
3. b_1_2·b_1_3² + b_1_2²·b_1_3 + b_2_8·b_1_2 + b_2_8·b_1_1 + b_2_8·a_1_0
4. b_2_8·b_1_1·b_1_3² + a_1_0·b_1_1²·b_1_3² + b_2_8·a_1_0·b_1_3²
      + b_2_8·a_1_0·b_1_2·b_1_3 + b_2_8·a_1_0·b_1_1·b_1_3
5. a_6_27·b_1_2
6. a_6_27·a_1_0
7. a_6_13·a_1_0
8. a_1_0·b_1_1⁴·b_1_3² + a_1_0·b_1_1⁵·b_1_3 + a_6_13·b_1_1 + a_6_27·b_1_1
9. a_6_13·b_1_3² + a_6_13·b_1_2·b_1_3 + a_6_13·b_1_1·b_1_3 + a_6_27·b_1_3²
      + a_6_27·b_1_1·b_1_3 + b_2_8·a_1_0·b_1_3⁵ + b_2_8·a_1_0·b_1_2⁴·b_1_3 + b_2_8·a_6_13
      + b_2_8²·a_1_0·b_1_3³
10. b_1_2·a_7_25 + a_6_13·b_1_2² + b_2_8·a_1_0·b_1_2⁴·b_1_3 + b_2_8³·a_1_0·b_1_1
11. b_2_8·a_6_27 + b_2_8³·a_1_0·b_1_1 + a_1_0·a_7_25
12. b_1_1·a_7_25 + a_6_27·b_1_1² + b_2_8·a_6_27
13. a_6_27²
14. a_6_27·a_6_13
15. a_6_13²
16. a_6_27·a_7_25
17. a_6_13·a_7_25 + b_2_8·a_1_0·b_1_3³·a_7_25
18. a_7_25²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 14.
• 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_43, a Duflot regular element of degree 8
  2. b_1_3⁴ + b_1_2³·b_1_3 + b_1_2⁴ + b_1_1²·b_1_3² + b_1_1⁴ + b_2_8·b_1_3²
        + b_2_8·b_1_2·b_1_3 + b_2_8·b_1_2² + b_2_8², an element of degree 4
  3. b_1_3², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 8, 11].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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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. b_1_3 → 0, an element of degree 1
5. b_2_8 → 0, an element of degree 2
6. a_6_27 → 0, an element of degree 6
7. a_6_13 → 0, an element of degree 6
8. a_7_25 → 0, an element of degree 7
9. c_8_43 → 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 → 0, an element of degree 1
4. b_1_3 → c_1_2, an element of degree 1
5. b_2_8 → 0, an element of degree 2
6. a_6_27 → 0, an element of degree 6
7. a_6_13 → 0, an element of degree 6
8. a_7_25 → 0, an element of degree 7
9. c_8_43 → 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

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 → 0, an element of degree 1
4. b_1_3 → c_1_1, an element of degree 1
5. b_2_8 → c_1_2², an element of degree 2
6. a_6_27 → 0, an element of degree 6
7. a_6_13 → 0, an element of degree 6
8. a_7_25 → 0, an element of degree 7
9. c_8_43 → c_1_1²·c_1_2⁶ + c_1_1⁴·c_1_2⁴ + 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

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. b_1_3 → c_1_2 + c_1_1, an element of degree 1
5. b_2_8 → c_1_2² + c_1_1·c_1_2, an element of degree 2
6. a_6_27 → 0, an element of degree 6
7. a_6_13 → 0, an element of degree 6
8. a_7_25 → 0, an element of degree 7
9. c_8_43 → c_1_1·c_1_2⁷ + c_1_1³·c_1_2⁵ + c_1_1⁵·c_1_2³ + 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