David J. Green

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Cohomology of group number 1745 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 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 3.
• The depth exceeds the Duflot bound, which is 2.
• The Poincaré series is

  ( − 1) · (t2  −  t  +  1) · (t2  +  t  +  1)2

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

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

About the group

Ring generators

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Ring generators

The cohomology ring has 8 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. b_2_8, an element of degree 2
6. c_2_9, a Duflot regular element of degree 2
7. b_5_45, an element of degree 5
8. c_8_99, a Duflot regular element of degree 8

About the group

Ring generators

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Ring relations

There are 10 minimal relations of maximal degree 10:

1. b_1_0·b_1_1
2. b_1_0·b_1_2
3. b_1_2·b_1_3² + b_1_2²·b_1_3 + b_1_1·b_1_3² + b_1_1·b_1_2·b_1_3 + b_1_1³
      + b_1_0·b_1_3²
4. b_2_8·b_1_0
5. b_1_1²·b_1_3² + b_1_1²·b_1_2·b_1_3 + b_1_1²·b_1_2² + b_1_1³·b_1_2 + b_1_1⁴
      + b_2_8·b_1_1·b_1_2 + b_2_8² + c_2_9·b_1_1²
6. b_2_8·b_1_1²·b_1_2 + b_2_8²·b_1_1 + c_2_9·b_1_1³
7. b_1_2·b_5_45 + b_1_2⁵·b_1_3 + b_1_1·b_1_3⁵ + b_1_1·b_1_2⁴·b_1_3 + b_1_1⁵·b_1_3
      + b_1_0·b_1_3⁵ + b_2_8·b_1_2³·b_1_3 + b_2_8·b_1_1·b_1_3³ + b_2_8·b_1_1·b_1_2³
      + b_2_8·b_1_1³·b_1_3 + b_2_8²·b_1_3² + b_2_8²·b_1_2² + b_2_8²·b_1_1·b_1_3
      + b_2_8²·b_1_1² + c_2_9·b_1_2³·b_1_3 + c_2_9·b_1_1·b_1_3³
      + c_2_9·b_1_1²·b_1_2² + c_2_9·b_1_1³·b_1_2 + c_2_9·b_1_0·b_1_3³
      + b_2_8·c_2_9·b_1_2·b_1_3 + b_2_8·c_2_9·b_1_1·b_1_2 + b_2_8²·c_2_9 + c_2_9²·b_1_1²
8. b_1_1·b_5_45 + b_1_1·b_1_3⁵ + b_1_1³·b_1_2²·b_1_3 + b_1_1⁵·b_1_3
      + b_2_8·b_1_1·b_1_3³ + b_2_8·b_1_1³·b_1_3 + b_2_8·b_1_1⁴ + b_2_8²·b_1_3²
      + b_2_8²·b_1_2·b_1_3 + b_2_8³ + c_2_9·b_1_1·b_1_3³ + c_2_9·b_1_1²·b_1_2·b_1_3
      + c_2_9·b_1_1²·b_1_2² + c_2_9·b_1_1³·b_1_3 + c_2_9·b_1_1³·b_1_2
      + b_2_8·c_2_9·b_1_1·b_1_3 + b_2_8·c_2_9·b_1_1·b_1_2 + b_2_8·c_2_9·b_1_1²
      + b_2_8²·c_2_9 + c_2_9²·b_1_1²
9. b_2_8·b_5_45 + b_2_8·b_1_3⁵ + b_2_8·b_1_1·b_1_3⁴ + b_2_8·b_1_1·b_1_2³·b_1_3
      + b_2_8·b_1_1⁴·b_1_3 + b_2_8·b_1_1⁵ + b_2_8²·b_1_3³ + b_2_8²·b_1_1·b_1_2·b_1_3
      + b_2_8²·b_1_1·b_1_2² + b_2_8²·b_1_1²·b_1_3 + b_2_8²·b_1_1³ + b_2_8³·b_1_1
      + c_2_9·b_1_1³·b_1_2·b_1_3 + c_2_9·b_1_1³·b_1_2² + b_2_8·c_2_9·b_1_3³
      + b_2_8·c_2_9·b_1_1·b_1_2·b_1_3 + b_2_8·c_2_9·b_1_1²·b_1_3 + b_2_8²·c_2_9·b_1_3
      + b_2_8²·c_2_9·b_1_1 + c_2_9²·b_1_1³
10. b_5_45² + b_1_3¹⁰ + b_1_1⁸·b_1_2·b_1_3 + b_1_1⁸·b_1_2² + b_1_1⁹·b_1_2 + b_1_1¹⁰
       + b_1_0⁴·b_1_3·b_5_45 + b_2_8²·b_1_2⁵·b_1_3 + b_2_8²·b_1_1⁶ + b_2_8³·b_1_3⁴
       + b_2_8³·b_1_1⁴ + c_8_99·b_1_0² + c_2_9·b_1_1⁶·b_1_2² + c_2_9·b_1_0³·b_5_45
       + b_2_8²·c_2_9·b_1_3⁴ + b_2_8²·c_2_9·b_1_2³·b_1_3
       + b_2_8²·c_2_9·b_1_1·b_1_2²·b_1_3 + b_2_8²·c_2_9·b_1_1⁴ + c_2_9²·b_1_3⁶
       + c_2_9²·b_1_1³·b_1_2²·b_1_3 + c_2_9²·b_1_1⁴·b_1_2·b_1_3
       + c_2_9²·b_1_1⁴·b_1_2² + c_2_9²·b_1_1⁵·b_1_2
       + b_2_8·c_2_9²·b_1_1·b_1_2²·b_1_3 + b_2_8²·c_2_9²·b_1_1² + b_2_8³·c_2_9²
       + c_2_9³·b_1_1²·b_1_2·b_1_3 + c_2_9³·b_1_1²·b_1_2² + c_2_9³·b_1_0³·b_1_3
       + b_2_8·c_2_9³·b_1_1·b_1_2 + b_2_8·c_2_9³·b_1_1² + b_2_8²·c_2_9³
       + c_2_9⁴·b_1_1²

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 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_9, a Duflot regular element of degree 2
  2. c_8_99, 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
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 9].
• 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 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. b_2_8 → 0, an element of degree 2
6. c_2_9 → c_1_0², an element of degree 2
7. b_5_45 → 0, an element of degree 5
8. c_8_99 → c_1_1⁸, 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. b_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. b_5_45 → c_1_1²·c_1_2³ + c_1_1⁴·c_1_2, an element of degree 5
8. c_8_99 → 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, 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. b_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. b_5_45 → 0, an element of degree 5
8. c_8_99 → c_1_1⁴·c_1_2⁴ + c_1_1⁸, 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 → 0, an element of degree 1
4. b_1_3 → c_1_2, an element of degree 1
5. b_2_8 → c_1_0·c_1_2, an element of degree 2
6. c_2_9 → c_1_0², an element of degree 2
7. b_5_45 → c_1_0·c_1_2⁴ + c_1_0²·c_1_2³ + c_1_0³·c_1_2², an element of degree 5
8. c_8_99 → c_1_1⁴·c_1_2⁴ + c_1_1⁸ + c_1_0²·c_1_2⁶ + c_1_0⁴·c_1_2⁴ + c_1_0⁵·c_1_2³
      + c_1_0⁶·c_1_2², 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. b_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. b_5_45 → c_1_2⁵ + c_1_0·c_1_2⁴ + c_1_0²·c_1_2³, an element of degree 5
8. c_8_99 → c_1_1⁴·c_1_2⁴ + c_1_1⁸ + c_1_0·c_1_2⁷ + c_1_0²·c_1_2⁶, 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 → 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. c_2_9 → c_1_0², an element of degree 2
7. b_5_45 → c_1_2⁵ + c_1_0²·c_1_2³, an element of degree 5
8. c_8_99 → c_1_1⁴·c_1_2⁴ + c_1_1⁸ + c_1_0²·c_1_2⁶ + c_1_0⁶·c_1_2², an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128