Cohomology of group number 2069 of order 128

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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 3 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 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t5  +  2·t4  +  2·t2  +  2·t  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-5,-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 4:

  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. b_1_3, an element of degree 1
  5. a_4_5, a nilpotent element of degree 4
  6. b_4_9, an element of degree 4
  7. b_4_10, an element of degree 4
  8. b_4_11, an element of degree 4
  9. c_4_12, a Duflot regular element of degree 4
  10. c_4_13, a Duflot regular element of degree 4

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Ring relations

There are 23 minimal relations of maximal degree 8:

  1. a_1_2·b_1_0
  2. b_1_1·b_1_3 + b_1_0·b_1_1 + b_1_02 + a_1_22
  3. b_1_0·b_1_12 + b_1_03 + a_1_2·b_1_12
  4. b_1_02·b_1_3 + b_1_02·b_1_1 + b_1_03 + a_1_22·b_1_3 + a_1_22·b_1_1
  5. a_4_5·a_1_2
  6. b_4_9·b_1_3 + a_4_5·b_1_3 + a_4_5·b_1_0
  7. b_4_9·b_1_1 + b_4_9·b_1_0 + a_4_5·b_1_0
  8. b_4_9·a_1_2
  9. b_4_10·b_1_0 + a_1_2·b_1_14 + a_4_5·b_1_1
  10. a_1_2·b_1_14 + b_4_10·a_1_2 + a_4_5·b_1_1 + a_4_5·b_1_0
  11. b_1_15 + b_4_11·b_1_1 + a_4_5·b_1_1
  12. b_1_05 + b_4_11·b_1_0 + a_1_2·b_1_14 + a_4_5·b_1_3 + a_4_5·b_1_0
  13. b_4_10·b_1_3 + b_4_11·a_1_2 + a_4_5·b_1_1 + a_4_5·b_1_0
  14. a_4_5·b_4_9 + a_4_52 + c_4_13·b_1_03·b_1_1 + c_4_13·b_1_04
  15. b_4_92 + a_4_52 + c_4_13·b_1_04
  16. b_4_10·a_1_2·b_1_13 + a_4_5·b_1_14 + a_4_5·b_1_04 + a_4_5·b_4_10
       + c_4_12·a_1_2·b_1_13
  17. b_4_9·b_4_10 + c_4_13·b_1_03·b_1_1 + c_4_13·b_1_04
  18. b_4_11·a_1_22·b_1_32 + a_4_52 + c_4_13·a_1_22·b_1_32 + c_4_12·a_1_22·b_1_32
  19. b_4_11·b_1_14 + b_4_11·b_1_04 + b_4_10·b_1_14 + b_4_102 + b_4_10·a_1_2·b_1_13
       + a_4_5·b_1_04 + a_4_52 + c_4_12·b_1_14 + c_4_12·b_1_04
  20. a_4_5·b_1_34 + a_4_5·b_1_14 + a_4_5·b_4_11 + a_4_52 + c_4_13·b_1_0·b_1_33
       + c_4_12·b_1_0·b_1_33
  21. b_4_11·b_1_34 + b_4_11·b_1_04 + b_4_112 + b_4_10·b_1_14 + b_4_102
       + b_4_11·a_1_2·b_1_33 + b_4_10·a_1_2·b_1_13 + a_4_5·b_1_34 + a_4_5·b_1_14
       + a_4_5·b_1_04 + a_4_52 + c_4_13·b_1_34 + c_4_12·b_1_34 + c_4_12·b_1_14
       + c_4_12·b_1_04 + c_4_13·a_1_22·b_1_32
  22. b_4_10·b_1_14 + b_4_10·b_4_11 + b_4_11·a_1_2·b_1_33 + b_4_10·a_1_2·b_1_13
       + a_4_5·b_1_14 + a_4_5·b_1_04 + c_4_13·a_1_2·b_1_33 + c_4_12·a_1_2·b_1_33
       + c_4_12·a_1_2·b_1_13 + c_4_13·a_1_22·b_1_32 + c_4_12·a_1_22·b_1_32
  23. b_4_9·b_1_04 + b_4_9·b_4_11 + a_4_5·b_1_34 + c_4_13·b_1_0·b_1_33
       + c_4_13·b_1_03·b_1_1 + c_4_13·b_1_04 + c_4_12·b_1_0·b_1_33

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 8.
  • 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_12, a Duflot regular element of degree 4
    2. c_4_13, a Duflot regular element of degree 4
    3. b_1_32 + b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 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_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. b_1_3 → 0, an element of degree 1
  5. a_4_5 → 0, an element of degree 4
  6. b_4_9 → 0, an element of degree 4
  7. b_4_10 → 0, an element of degree 4
  8. b_4_11 → 0, an element of degree 4
  9. c_4_12 → c_1_04, an element of degree 4
  10. c_4_13 → c_1_14 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. 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. b_1_3 → c_1_2, an element of degree 1
  5. a_4_5 → 0, an element of degree 4
  6. b_4_9 → 0, an element of degree 4
  7. b_4_10 → 0, an element of degree 4
  8. b_4_11 → c_1_12·c_1_22, an element of degree 4
  9. c_4_12 → c_1_12·c_1_22 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  10. c_4_13 → c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4

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

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → c_1_2, an element of degree 1
  3. b_1_1 → c_1_2, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. a_4_5 → 0, an element of degree 4
  6. b_4_9 → c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  7. b_4_10 → 0, an element of degree 4
  8. b_4_11 → c_1_24, an element of degree 4
  9. c_4_12 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  10. c_4_13 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. 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 → c_1_2, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. a_4_5 → 0, an element of degree 4
  6. b_4_9 → 0, an element of degree 4
  7. b_4_10 → c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  8. b_4_11 → c_1_24, an element of degree 4
  9. c_4_12 → c_1_24 + c_1_0·c_1_23 + c_1_04, an element of degree 4
  10. c_4_13 → c_1_24 + c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4

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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