David J. Green

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Cohomology of group number 2003 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 2.
• It has 5 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 coincides with the Duflot bound.
• The Poincaré series is

  ( − 1) · (t7  +  2·t6  +  2·t5  +  t3  +  2·t2  +  2·t  +  1)

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

• The a-invariants are -∞,-∞,-6,-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. 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_1_3, an element of degree 1
5. c_2_8, a Duflot regular element of degree 2
6. b_5_26, an element of degree 5
7. b_5_27, an element of degree 5
8. b_5_28, an element of degree 5
9. c_8_61, a Duflot regular element of degree 8

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Ring relations

There are 16 minimal relations of maximal degree 10:

1. b_1_0·b_1_2
2. b_1_1·b_1_3 + b_1_0·b_1_1
3. b_1_3³ + b_1_2²·b_1_3 + b_1_1·b_1_2² + b_1_1³ + b_1_0·b_1_3²
4. b_1_0²·b_1_1³
5. b_1_1·b_5_26 + c_2_8·b_1_1⁴ + c_2_8·b_1_0·b_1_1³ + c_2_8²·b_1_1²
6. b_1_3·b_5_27 + b_1_2·b_5_26 + c_2_8·b_1_2³·b_1_3 + c_2_8·b_1_1³·b_1_2
      + c_2_8²·b_1_1·b_1_2
7. b_1_0·b_5_27
8. b_1_2·b_5_28 + b_1_1·b_5_27 + b_1_1·b_1_2⁵ + b_1_1⁵·b_1_2 + c_2_8·b_1_1·b_1_2³
      + c_2_8·b_1_1³·b_1_2
9. b_1_3·b_5_28 + b_1_0·b_5_28 + c_2_8·b_1_0·b_1_1³
10. b_1_3²·b_5_26 + b_1_2²·b_5_26 + b_1_1·b_1_2·b_5_27 + b_1_1²·b_5_28
       + c_2_8·b_1_1·b_1_2⁴ + c_2_8²·b_1_1·b_1_2² + c_2_8²·b_1_0²·b_1_1
11. b_5_26·b_5_28 + c_2_8·b_1_1³·b_5_28 + c_2_8²·b_1_1·b_5_28 + c_2_8²·b_1_0⁴·b_1_1²
12. b_5_27² + b_5_26·b_5_28 + b_1_2⁴·b_1_3·b_5_26 + b_1_2⁸·b_1_3² + b_1_2¹⁰
       + b_1_1⁵·b_5_28 + b_1_1¹⁰ + c_8_61·b_1_2² + c_2_8·b_1_2³·b_5_27 + c_2_8·b_1_2⁸
       + c_2_8·b_1_1⁸ + c_2_8²·b_1_1·b_5_28 + c_2_8²·b_1_0⁴·b_1_1² + c_2_8⁴·b_1_2²
13. b_5_27·b_5_28 + b_1_1·b_1_2⁴·b_5_27 + b_1_1·b_1_2⁹ + b_1_1⁹·b_1_2
       + c_8_61·b_1_1·b_1_2 + c_2_8·b_1_1·b_1_2⁷ + c_2_8·b_1_1⁷·b_1_2 + c_2_8⁴·b_1_1·b_1_2
14. b_5_26·b_5_27 + b_1_2⁵·b_5_26 + b_1_1·b_1_2⁴·b_5_27 + b_1_1·b_1_2⁹ + b_1_1⁵·b_5_27
       + b_1_1⁹·b_1_2 + c_8_61·b_1_2·b_1_3 + c_2_8·b_1_2⁷·b_1_3 + c_2_8·b_1_1·b_1_2⁷
       + c_2_8·b_1_1³·b_5_27 + c_2_8²·b_1_1·b_5_27 + c_2_8²·b_1_1·b_1_2⁵
       + c_2_8⁴·b_1_2·b_1_3
15. b_5_26² + b_1_2⁴·b_1_3·b_5_26 + c_8_61·b_1_3² + c_8_61·b_1_0²
       + c_2_8·b_1_2²·b_1_3·b_5_26 + c_2_8·b_1_2⁶·b_1_3² + c_2_8·b_1_0·b_1_1²·b_5_28
       + c_2_8·b_1_0³·b_5_26 + c_2_8²·b_1_1⁶ + c_2_8³·b_1_0³·b_1_1 + c_2_8⁴·b_1_3²
       + c_2_8⁴·b_1_1² + c_2_8⁴·b_1_0²
16. b_5_28² + b_1_1⁵·b_5_28 + b_1_0³·b_1_1²·b_5_28 + b_1_0⁴·b_1_1·b_5_28
       + c_8_61·b_1_1² + c_2_8·b_1_0·b_1_1²·b_5_28 + c_2_8·b_1_0⁷·b_1_1
       + c_2_8²·b_1_0⁴·b_1_3² + c_2_8⁴·b_1_1²

About the group

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 10.
• 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_8, a Duflot regular element of degree 2
  2. c_8_61, a Duflot regular element of degree 8
  3. b_1_2² + b_1_0², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 9].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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. 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_1_3 → 0, an element of degree 1
5. c_2_8 → c_1_0², an element of degree 2
6. b_5_26 → 0, an element of degree 5
7. b_5_27 → 0, an element of degree 5
8. b_5_28 → 0, an element of degree 5
9. c_8_61 → c_1_1⁸ + c_1_0⁸, an element of degree 8

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_1_3 → 0, an element of degree 1
5. c_2_8 → c_1_0·c_1_2 + c_1_0², an element of degree 2
6. b_5_26 → c_1_1²·c_1_2³ + c_1_1⁴·c_1_2, an element of degree 5
7. b_5_27 → 0, an element of degree 5
8. b_5_28 → 0, an element of degree 5
9. c_8_61 → c_1_1⁴·c_1_2⁴ + c_1_1⁸ + c_1_0·c_1_1²·c_1_2⁵ + c_1_0·c_1_1⁴·c_1_2³
      + 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_0⁸, an element of degree 8

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

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 → c_1_2, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. c_2_8 → c_1_0·c_1_2 + c_1_0², an element of degree 2
6. b_5_26 → c_1_0·c_1_2⁴ + c_1_0⁴·c_1_2, an element of degree 5
7. b_5_27 → c_1_1²·c_1_2³ + c_1_1⁴·c_1_2 + c_1_0·c_1_2⁴ + c_1_0²·c_1_2³, an element of degree 5
8. b_5_28 → c_1_1²·c_1_2³ + c_1_1⁴·c_1_2 + c_1_0·c_1_2⁴ + c_1_0²·c_1_2³, an element of degree 5
9. c_8_61 → c_1_1²·c_1_2⁶ + c_1_1⁸ + c_1_0·c_1_2⁷ + c_1_0⁸, an element of degree 8

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

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_1_3 → c_1_2, an element of degree 1
5. c_2_8 → c_1_0·c_1_2 + c_1_0², an element of degree 2
6. b_5_26 → c_1_1²·c_1_2³ + c_1_1⁴·c_1_2, an element of degree 5
7. b_5_27 → c_1_1²·c_1_2³ + c_1_1⁴·c_1_2 + c_1_0·c_1_2⁴ + c_1_0²·c_1_2³, an element of degree 5
8. b_5_28 → 0, an element of degree 5
9. c_8_61 → c_1_1²·c_1_2⁶ + c_1_1⁸ + c_1_0·c_1_2⁷ + c_1_0·c_1_1²·c_1_2⁵
      + c_1_0·c_1_1⁴·c_1_2³ + c_1_0²·c_1_2⁶ + 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_0⁸, an element of degree 8

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

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_1_3 → 0, an element of degree 1
5. c_2_8 → c_1_0·c_1_2 + c_1_0², an element of degree 2
6. b_5_26 → 0, an element of degree 5
7. b_5_27 → c_1_1²·c_1_2³ + c_1_1⁴·c_1_2, an element of degree 5
8. b_5_28 → 0, an element of degree 5
9. c_8_61 → c_1_2⁸ + c_1_1⁴·c_1_2⁴ + c_1_1⁸ + c_1_0·c_1_2⁷ + c_1_0·c_1_1²·c_1_2⁵
      + c_1_0·c_1_1⁴·c_1_2³ + c_1_0²·c_1_2⁶ + 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_0⁸, an element of degree 8

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_1_3 → c_1_2, an element of degree 1
5. c_2_8 → c_1_0·c_1_2 + c_1_0², an element of degree 2
6. b_5_26 → 0, an element of degree 5
7. b_5_27 → 0, an element of degree 5
8. b_5_28 → c_1_0·c_1_2⁴ + c_1_0²·c_1_2³, an element of degree 5
9. c_8_61 → c_1_2⁸ + c_1_1⁴·c_1_2⁴ + c_1_1⁸ + c_1_0⁴·c_1_2⁴ + 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