David J. Green

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Cohomology of group number 8935 of order 256

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General information on the group

• The group is also known as Syl2(S4(4)), the Sylow 2-group of Symplectic group S4(4).
• The group has 4 minimal generators and exponent 4.
• It is non-abelian.
• It has p-Rank 6.
• Its center has rank 4.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 6.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 6 and depth 4.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  t4  −  t3  +  2·t2  +  t  +  1

  (t  +  1)3 · (t  −  1)6

• The a-invariants are -∞,-∞,-∞,-∞,-5,-8,-6. They were obtained using the second power of the 1st, the second power of the 2nd, the second power of the 3rd, the second power of the 4th, the second power of the 5th, and the 6th filter regular parameter of the Benson test.

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Ring generators

The cohomology ring has 12 minimal generators of maximal degree 2:

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_6, an element of degree 2
6. b_2_7, an element of degree 2
7. b_2_8, an element of degree 2
8. b_2_9, an element of degree 2
9. c_2_10, a Duflot regular element of degree 2
10. c_2_11, a Duflot regular element of degree 2
11. c_2_12, a Duflot regular element of degree 2
12. c_2_13, a Duflot regular element of degree 2

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Ring relations

There are 19 minimal relations of maximal degree 4:

1. b_1_0·b_1_2
2. b_1_1·b_1_2
3. b_1_0·b_1_3 + b_1_0²
4. b_1_1·b_1_3 + b_1_0·b_1_1
5. b_2_6·b_1_2
6. b_2_7·b_1_2 + b_2_6·b_1_3 + b_2_6·b_1_0
7. b_2_7·b_1_3 + b_2_7·b_1_0
8. b_2_8·b_1_0
9. b_2_8·b_1_1 + b_2_6·b_1_3 + b_2_6·b_1_0
10. b_2_9·b_1_0 + b_2_6·b_1_3 + b_2_6·b_1_0
11. b_2_9·b_1_1
12. b_2_6·b_1_0·b_1_1 + b_2_6² + c_2_12·b_1_0² + c_2_10·b_1_1²
13. b_2_7·b_1_0·b_1_1 + b_2_7² + c_2_13·b_1_0² + c_2_11·b_1_1²
14. b_2_8·b_1_2·b_1_3 + b_2_8² + c_2_11·b_1_2² + c_2_10·b_1_3² + c_2_10·b_1_0²
15. b_2_7·b_2_8
16. b_2_6·b_2_8
17. b_2_9·b_1_2·b_1_3 + b_2_9² + c_2_13·b_1_2² + c_2_12·b_1_3² + c_2_12·b_1_0²
18. b_2_7·b_2_9
19. b_2_6·b_2_9

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 6.
• However, the last relation was already found in degree 4 and the last generator in degree 2.
• 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. c_2_13, a Duflot regular element of degree 2
  5. b_1_3² + b_1_2·b_1_3 + b_1_2² + b_1_1² + b_1_0·b_1_1, an element of degree 2
  6. b_1_2·b_1_3² + b_1_2²·b_1_3 + b_1_0·b_1_1² + b_1_0²·b_1_1, an element of degree 3
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 3, 3, 7].
• 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 4 terms of the above HSOP, together with 2 elements of degree 2.

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 4

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_6 → 0, an element of degree 2
6. b_2_7 → 0, an element of degree 2
7. b_2_8 → 0, an element of degree 2
8. b_2_9 → 0, an element of degree 2
9. c_2_10 → c_1_1², an element of degree 2
10. c_2_11 → c_1_3², an element of degree 2
11. c_2_12 → c_1_2², an element of degree 2
12. c_2_13 → c_1_0², an element of degree 2

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

1. b_1_0 → c_1_5, an element of degree 1
2. b_1_1 → c_1_4, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. b_1_3 → c_1_5, an element of degree 1
5. b_2_6 → c_1_2·c_1_5 + c_1_1·c_1_4, an element of degree 2
6. b_2_7 → c_1_4·c_1_5 + c_1_3·c_1_4 + c_1_0·c_1_5, an element of degree 2
7. b_2_8 → 0, an element of degree 2
8. b_2_9 → 0, an element of degree 2
9. c_2_10 → c_1_1·c_1_5 + c_1_1², an element of degree 2
10. c_2_11 → c_1_3·c_1_5 + c_1_3², an element of degree 2
11. c_2_12 → c_1_2·c_1_4 + c_1_2², an element of degree 2
12. c_2_13 → c_1_0·c_1_4 + c_1_0², an element of degree 2

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 → 0, 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_6 → 0, an element of degree 2
6. b_2_7 → 0, an element of degree 2
7. b_2_8 → c_1_3·c_1_4 + c_1_1·c_1_5, an element of degree 2
8. b_2_9 → c_1_2·c_1_5 + c_1_0·c_1_4, an element of degree 2
9. c_2_10 → c_1_1·c_1_4 + c_1_1², an element of degree 2
10. c_2_11 → c_1_3·c_1_5 + c_1_3², an element of degree 2
11. c_2_12 → c_1_2·c_1_4 + c_1_2², an element of degree 2
12. c_2_13 → c_1_0·c_1_5 + c_1_0², an element of degree 2

About the group

Ring generators

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Completion information

Restriction maps

Back to groups of order 256