David J. Green

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Cohomology of group number 1819 of order 128

About the group

Ring generators

Ring relations

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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 3.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 3.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  (2) · (t3  +  1/2·t  +  1/2)

  (t  +  1) · (t  −  1)4 · (t2  +  1)2

• The a-invariants are -∞,-∞,-∞,-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 5:

1. a_1_2, a nilpotent element of degree 1
2. b_1_0, an element of degree 1
3. b_1_1, an element of degree 1
4. c_1_3, a Duflot regular element of degree 1
5. a_3_4, a nilpotent element of degree 3
6. b_3_10, an element of degree 3
7. b_3_12, an element of degree 3
8. b_3_13, an element of degree 3
9. c_4_22, a Duflot regular element of degree 4
10. c_4_23, a Duflot regular element of degree 4
11. b_5_37, an element of degree 5

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Ring relations

There are 27 minimal relations of maximal degree 10:

1. a_1_2·b_1_0
2. b_1_0·b_1_1 + a_1_2²
3. a_1_2·b_1_1²
4. a_1_2²·b_1_1
5. b_1_0·a_3_4
6. b_1_1·b_3_10 + b_1_1⁴ + a_1_2·a_3_4
7. a_1_2·b_3_10
8. a_1_2·b_3_12
9. b_1_0·b_3_13 + b_1_0·b_3_12 + b_1_0·b_3_10 + a_1_2·a_3_4
10. b_1_1·b_3_12 + b_1_1·a_3_4 + a_1_2·b_3_13
11. a_1_2·b_1_1·b_3_13
12. a_3_4·b_3_10 + b_1_1³·a_3_4
13. a_3_4·b_3_12
14. b_3_10·b_3_13 + b_3_10·b_3_12 + b_3_10² + b_1_1³·b_3_13 + b_1_1⁶ + b_1_1³·a_3_4
       + a_3_4²
15. b_3_12·b_3_13 + b_3_12² + b_3_10·b_3_12 + a_3_4·b_3_13 + b_1_1³·a_3_4
       + c_4_22·a_1_2·b_1_1
16. b_3_10² + b_1_1⁶ + c_4_22·b_1_0²
17. a_3_4² + c_4_22·a_1_2²
18. b_3_12² + c_4_23·b_1_0²
19. b_3_13² + b_3_12² + b_3_10² + b_1_1³·a_3_4 + c_4_22·b_1_1² + c_4_23·a_1_2²
20. b_1_1·b_5_37 + b_1_1⁶ + a_3_4·b_3_13 + b_1_1³·a_3_4 + a_3_4² + c_4_23·b_1_1²
       + c_4_22·b_1_1² + c_4_23·a_1_2·b_1_1
21. b_3_13² + b_3_12² + b_3_10·b_3_12 + b_1_1⁶ + b_1_0·b_5_37 + a_3_4² + c_4_22·b_1_1²
22. b_3_13² + b_3_12·b_3_13 + b_3_10·b_3_12 + b_3_10² + a_3_4·b_3_13 + a_1_2·b_5_37
       + a_3_4² + c_4_22·b_1_1² + c_4_23·a_1_2·b_1_1
23. b_3_12·b_5_37 + b_1_1⁵·a_3_4 + c_4_23·b_1_0·b_3_10 + c_4_22·b_1_0·b_3_12
       + c_4_23·b_1_1·a_3_4 + c_4_23·a_1_2·b_3_13 + c_4_22·b_1_1·a_3_4 + c_4_22·a_1_2·b_3_13
24. b_3_13·b_5_37 + b_1_1⁵·b_3_13 + b_1_1²·a_3_4·b_3_13 + b_1_1⁵·a_3_4
       + c_4_23·b_1_1·b_3_13 + c_4_23·b_1_0·b_3_10 + c_4_22·b_1_1·b_3_13 + c_4_22·b_1_0·b_3_10
       + c_4_23·a_1_2·b_3_13 + c_4_22·b_1_1·a_3_4 + c_4_22·a_1_2·a_3_4
25. b_3_10·b_5_37 + b_1_1⁸ + b_1_1²·a_3_4·b_3_13 + b_1_1⁵·a_3_4 + c_4_23·b_1_1⁴
       + c_4_22·b_1_1⁴ + c_4_22·b_1_0·b_3_12 + c_4_22·b_1_0·b_3_10 + c_4_23·a_1_2·a_3_4
       + c_4_22·a_1_2·a_3_4
26. a_3_4·b_5_37 + b_1_1⁵·a_3_4 + c_4_23·b_1_1·a_3_4 + c_4_22·b_1_1·a_3_4
       + c_4_23·a_1_2·a_3_4 + c_4_22·a_1_2·a_3_4
27. b_5_37² + b_1_1¹⁰ + c_4_23²·b_1_1² + c_4_22·c_4_23·b_1_0² + c_4_22²·b_1_1²
       + c_4_22²·b_1_0² + c_4_23²·a_1_2² + c_4_22²·a_1_2²

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_1_3, a Duflot regular element of degree 1
  2. c_4_22, a Duflot regular element of degree 4
  3. c_4_23, a Duflot regular element of degree 4
  4. b_1_1² + b_1_0², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 7].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

About the group

Ring generators

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 3

1. a_1_2 → 0, an element of degree 1
2. b_1_0 → 0, an element of degree 1
3. b_1_1 → 0, an element of degree 1
4. c_1_3 → c_1_0, an element of degree 1
5. a_3_4 → 0, an element of degree 3
6. b_3_10 → 0, an element of degree 3
7. b_3_12 → 0, an element of degree 3
8. b_3_13 → 0, an element of degree 3
9. c_4_22 → c_1_1⁴, an element of degree 4
10. c_4_23 → c_1_2⁴ + c_1_1⁴, an element of degree 4
11. b_5_37 → 0, an element of degree 5

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

1. a_1_2 → 0, an element of degree 1
2. b_1_0 → c_1_3, an element of degree 1
3. b_1_1 → 0, an element of degree 1
4. c_1_3 → c_1_0, an element of degree 1
5. a_3_4 → 0, an element of degree 3
6. b_3_10 → c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
7. b_3_12 → c_1_2·c_1_3² + c_1_2²·c_1_3 + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
8. b_3_13 → c_1_2·c_1_3² + c_1_2²·c_1_3, an element of degree 3
9. c_4_22 → c_1_1²·c_1_3² + c_1_1⁴, an element of degree 4
10. c_4_23 → c_1_2²·c_1_3² + c_1_2⁴ + c_1_1²·c_1_3² + c_1_1⁴, an element of degree 4
11. b_5_37 → c_1_1·c_1_2·c_1_3³ + c_1_1·c_1_2²·c_1_3² + c_1_1²·c_1_2·c_1_3²
       + c_1_1²·c_1_2²·c_1_3, an element of degree 5

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

1. a_1_2 → 0, an element of degree 1
2. b_1_0 → 0, an element of degree 1
3. b_1_1 → c_1_3, an element of degree 1
4. c_1_3 → c_1_0, an element of degree 1
5. a_3_4 → 0, an element of degree 3
6. b_3_10 → c_1_3³, an element of degree 3
7. b_3_12 → 0, an element of degree 3
8. b_3_13 → c_1_3³ + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
9. c_4_22 → c_1_1²·c_1_3² + c_1_1⁴, an element of degree 4
10. c_4_23 → c_1_2²·c_1_3² + c_1_2⁴ + c_1_1·c_1_3³ + c_1_1⁴, an element of degree 4
11. b_5_37 → c_1_3⁵ + c_1_2²·c_1_3³ + c_1_2⁴·c_1_3 + c_1_1·c_1_3⁴ + c_1_1²·c_1_3³, an element of degree 5

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128