David J. Green

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Cohomology of group number 1578 of order 128

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General information on the group

• The group is also known as V8wrC2, the Wreath product V_8 wr C_2.
• The group has 4 minimal generators and exponent 4.
• It is non-abelian.
• It has p-Rank 6.
• Its center has rank 3.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 4 and 6, respectively.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 6 and depth 4.
• The depth exceeds the Duflot bound, which is 3.
• The Poincaré series is

  t2  +  1

  (t  +  1)2 · (t  −  1)6

• The a-invariants are -∞,-∞,-∞,-∞,-6,-6,-6. 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 3:

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_7, an element of degree 2
6. b_2_8, an element of degree 2
7. b_2_9, an element of degree 2
8. c_2_10, a Duflot regular element of degree 2
9. c_2_11, a Duflot regular element of degree 2
10. c_2_12, a Duflot regular element of degree 2
11. b_3_31, an element of degree 3

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Ring relations

There are 18 minimal relations of maximal degree 6:

1. b_1_0·b_1_1
2. b_1_0·b_1_2
3. b_1_0·b_1_3
4. b_2_7·b_1_0
5. b_2_8·b_1_0
6. b_2_9·b_1_0
7. b_2_9·b_1_1 + b_2_8·b_1_2 + b_2_7·b_1_3
8. b_2_7·b_1_1·b_1_2 + b_2_7² + c_2_11·b_1_1² + c_2_10·b_1_2²
9. b_2_8·b_1_1·b_1_3 + b_2_8² + c_2_12·b_1_1² + c_2_10·b_1_3²
10. b_2_9·b_1_2·b_1_3 + b_2_9² + c_2_12·b_1_2² + c_2_11·b_1_3²
11. b_1_2·b_3_31 + b_2_7·b_2_9 + c_2_11·b_1_1·b_1_3
12. b_1_3·b_3_31 + b_2_8·b_1_2·b_1_3 + b_2_8·b_2_9 + b_2_7·b_1_3² + c_2_12·b_1_1·b_1_2
13. b_1_0·b_3_31
14. b_1_1·b_3_31 + b_2_7·b_1_1·b_1_3 + b_2_7·b_2_8 + c_2_10·b_1_2·b_1_3
15. b_2_9·b_3_31 + b_2_7·b_2_9·b_1_3 + b_2_8·c_2_11·b_1_3 + b_2_7·c_2_12·b_1_2
16. b_2_8·b_3_31 + b_2_9·c_2_10·b_1_3 + b_2_7·c_2_12·b_1_1
17. b_2_7·b_3_31 + b_2_7·b_2_8·b_1_2 + b_2_7²·b_1_3 + b_2_9·c_2_10·b_1_2
       + b_2_8·c_2_11·b_1_1
18. b_3_31² + b_2_7·b_2_8·b_1_2·b_1_3 + b_2_7²·b_1_3² + b_2_9²·c_2_10 + b_2_8²·c_2_11
       + b_2_7²·c_2_12 + c_2_11·c_2_12·b_1_1² + c_2_10·c_2_12·b_1_2²
       + c_2_10·c_2_11·b_1_3²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 6.
• 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_10, a Duflot regular element of degree 2
  2. c_2_11, a Duflot regular element of degree 2
  3. c_2_12, a Duflot regular element of degree 2
  4. b_1_3⁴ + b_1_2²·b_1_3² + b_1_2⁴ + b_1_1·b_1_2·b_1_3² + b_1_1·b_1_2²·b_1_3
        + b_1_1²·b_1_3² + b_1_1²·b_1_2·b_1_3 + b_1_1²·b_1_2² + b_1_1⁴ + b_1_0⁴, an element of degree 4
  5. b_1_2²·b_1_3⁴ + b_1_2⁴·b_1_3² + b_1_1·b_1_2·b_1_3⁴ + b_1_1·b_1_2⁴·b_1_3
        + b_1_1²·b_1_3⁴ + b_1_1²·b_1_2²·b_1_3² + b_1_1²·b_1_2⁴ + b_1_1⁴·b_1_3²
        + b_1_1⁴·b_1_2·b_1_3 + b_1_1⁴·b_1_2², an element of degree 6
  6. b_1_3², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 4, 10, 12].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -6, -6].
• We found that there exists some filter regular HSOP formed by the first 3 terms of the above HSOP, together with 3 elements of degree 2.

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is 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 → 0, an element of degree 1
5. b_2_7 → 0, an element of degree 2
6. b_2_8 → 0, an element of degree 2
7. b_2_9 → 0, an element of degree 2
8. c_2_10 → c_1_0², an element of degree 2
9. c_2_11 → c_1_1², an element of degree 2
10. c_2_12 → c_1_2², an element of degree 2
11. b_3_31 → 0, an element of degree 3

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

1. b_1_0 → c_1_3, 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_7 → 0, an element of degree 2
6. b_2_8 → 0, an element of degree 2
7. b_2_9 → 0, an element of degree 2
8. c_2_10 → c_1_0·c_1_3 + c_1_0², an element of degree 2
9. c_2_11 → c_1_1·c_1_3 + c_1_1², an element of degree 2
10. c_2_12 → c_1_2·c_1_3 + c_1_2², an element of degree 2
11. b_3_31 → 0, an element of degree 3

Restriction map to a maximal el. ab. subgp. of rank 6

1. b_1_0 → 0, an element of degree 1
2. b_1_1 → c_1_3, an element of degree 1
3. b_1_2 → c_1_4, an element of degree 1
4. b_1_3 → c_1_5, an element of degree 1
5. b_2_7 → c_1_1·c_1_3 + c_1_0·c_1_4, an element of degree 2
6. b_2_8 → c_1_2·c_1_3 + c_1_0·c_1_5, an element of degree 2
7. b_2_9 → c_1_2·c_1_4 + c_1_1·c_1_5, an element of degree 2
8. c_2_10 → c_1_0·c_1_3 + c_1_0², an element of degree 2
9. c_2_11 → c_1_1·c_1_4 + c_1_1², an element of degree 2
10. c_2_12 → c_1_2·c_1_5 + c_1_2², an element of degree 2
11. b_3_31 → c_1_1·c_1_3·c_1_5 + c_1_1·c_1_2·c_1_3 + c_1_0·c_1_2·c_1_4 + c_1_0·c_1_1·c_1_5, an element of degree 3

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128