Cohomology of group number 284 of order 128

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General information on the group

  • The group has 3 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is 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
    (t  −  1)3 · (t2  +  1)
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 10 minimal generators of maximal degree 5:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. b_1_2, an element of degree 1
  4. a_2_4, a nilpotent element of degree 2
  5. a_3_5, a nilpotent element of degree 3
  6. a_3_6, a nilpotent element of degree 3
  7. b_4_7, an element of degree 4
  8. c_4_8, a Duflot regular element of degree 4
  9. c_4_9, a Duflot regular element of degree 4
  10. a_5_6, a nilpotent element of degree 5

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Ring relations

There are 27 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_1_13
  4. a_1_1·b_1_22
  5. a_2_4·a_1_0
  6. a_2_42 + a_2_4·a_1_12
  7. a_1_1·a_3_5
  8. a_1_0·a_3_5 + a_2_4·a_1_12
  9. a_1_1·a_3_6 + a_2_4·a_1_12
  10. a_1_0·b_1_23 + a_2_4·b_1_22 + a_1_0·a_3_6 + a_2_4·a_1_12
  11. a_2_4·a_3_5
  12. a_1_0·b_1_2·a_3_6 + a_2_4·a_3_6
  13. b_4_7·a_1_1
  14. b_1_22·a_3_5 + b_4_7·a_1_0
  15. a_3_52
  16. a_3_62 + a_2_4·b_1_2·a_3_6
  17. b_4_7·a_1_0·b_1_2 + a_2_4·b_4_7 + a_3_5·a_3_6
  18. a_1_1·a_5_6 + c_4_9·a_1_12
  19. a_3_5·a_3_6 + a_1_0·a_5_6
  20. b_4_7·a_3_5 + a_2_4·b_1_25 + a_2_4·b_1_22·a_3_6 + c_4_8·a_1_0·b_1_22
  21. b_1_22·a_5_6 + b_4_7·a_3_6
  22. a_1_0·b_1_2·a_5_6 + a_2_4·a_5_6 + a_2_4·c_4_9·a_1_1
  23. b_1_28 + b_4_72 + c_4_8·b_1_24
  24. a_3_5·a_5_6 + a_2_4·b_1_23·a_3_6 + c_4_8·a_1_0·a_3_6
  25. a_3_6·a_5_6 + a_2_4·b_1_2·a_5_6 + a_2_4·c_4_9·a_1_1·b_1_2 + a_2_4·c_4_8·a_1_12
  26. b_1_26·a_3_6 + b_4_7·a_5_6 + c_4_8·b_1_22·a_3_6
  27. a_5_62 + a_2_4·b_1_25·a_3_6 + a_2_4·c_4_8·b_1_2·a_3_6 + c_4_92·a_1_12

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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_4_8, a Duflot regular element of degree 4
    2. c_4_9, a Duflot regular element of degree 4
    3. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 7].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_3_5 → 0, an element of degree 3
  6. a_3_6 → 0, an element of degree 3
  7. b_4_7 → 0, an element of degree 4
  8. c_4_8 → c_1_04, an element of degree 4
  9. c_4_9 → c_1_14, an element of degree 4
  10. a_5_6 → 0, an element of degree 5

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_3_5 → 0, an element of degree 3
  6. a_3_6 → 0, an element of degree 3
  7. b_4_7 → c_1_24 + c_1_02·c_1_22, an element of degree 4
  8. c_4_8 → c_1_04, an element of degree 4
  9. c_4_9 → c_1_12·c_1_22 + c_1_14, an element of degree 4
  10. a_5_6 → 0, an element of degree 5

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Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2

Last change: 25.08.2009