Cohomology of group number 698 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) · (t5  +  t2  +  1)
    (t  −  1)3 · (t2  +  1) · (t4  +  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 11 minimal generators of maximal degree 8:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_1_2, a nilpotent element of degree 1
  4. a_2_3, a nilpotent element of degree 2
  5. b_2_4, an element of degree 2
  6. c_2_5, a Duflot regular element of degree 2
  7. a_3_9, a nilpotent element of degree 3
  8. a_5_15, a nilpotent element of degree 5
  9. b_5_17, an element of degree 5
  10. a_6_21, a nilpotent element of degree 6
  11. c_8_37, a Duflot regular element of degree 8

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Ring relations

There are 28 minimal relations of maximal degree 12:

  1. a_1_12 + a_1_02
  2. a_1_0·a_1_1 + a_1_02
  3. a_1_22 + a_1_0·a_1_2
  4. a_2_3·a_1_1 + a_2_3·a_1_0
  5. b_2_4·a_1_1 + b_2_4·a_1_0 + a_1_03
  6. a_2_32 + a_2_3·a_1_02
  7. a_2_3·b_2_4 + a_1_1·a_3_9 + a_2_3·a_1_02
  8. a_2_3·b_2_4 + a_1_0·a_3_9
  9. b_2_4·a_1_03
  10. a_2_3·a_3_9 + a_1_02·a_3_9
  11. a_3_92 + b_2_4·a_1_0·a_3_9 + b_2_42·a_1_02 + a_1_03·a_3_9
  12. a_1_1·a_5_15 + b_2_42·a_1_02
  13. a_1_0·a_5_15 + b_2_42·a_1_02 + a_1_03·a_3_9
  14. a_1_1·b_5_17 + a_1_0·b_5_17
  15. a_2_3·a_5_15 + b_2_4·a_1_02·a_3_9
  16. b_2_4·a_5_15 + b_2_43·a_1_0 + a_1_02·b_5_17
  17. b_2_4·a_5_15 + b_2_43·a_1_0 + a_2_3·b_5_17 + a_6_21·a_1_1 + b_2_4·a_1_02·a_3_9
  18. b_2_4·a_5_15 + b_2_43·a_1_0 + a_2_3·b_5_17 + a_6_21·a_1_0 + b_2_4·a_1_02·a_3_9
  19. a_3_9·a_5_15 + b_2_42·a_1_0·a_3_9 + a_6_21·a_1_02
  20. a_3_9·b_5_17 + b_2_4·a_1_0·b_5_17 + b_2_4·a_6_21 + a_3_9·a_5_15 + b_2_43·a_1_02
  21. a_2_3·a_6_21
  22. a_6_21·a_3_9 + b_2_4·a_1_02·b_5_17
  23. a_5_152 + b_2_44·a_1_02
  24. a_5_15·b_5_17 + b_2_42·a_1_0·b_5_17 + b_2_44·a_1_02
  25. b_5_172 + b_2_45 + b_2_42·a_1_0·b_5_17 + c_8_37·a_1_02
  26. a_6_21·a_5_15 + b_2_42·a_6_21·a_1_0 + b_2_43·a_1_02·a_3_9
  27. a_6_21·b_5_17 + b_2_44·a_3_9 + b_2_45·a_1_0 + b_2_43·a_1_02·a_3_9
       + a_2_3·c_8_37·a_1_0 + c_8_37·a_1_03
  28. a_6_212 + b_2_44·a_1_0·a_3_9 + b_2_42·a_6_21·a_1_02 + a_2_3·c_8_37·a_1_02

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 12.
  • 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_5, a Duflot regular element of degree 2
    2. c_8_37, a Duflot regular element of degree 8
    3. b_2_4, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 9].
  • 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. a_1_2 → 0, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. c_2_5 → c_1_02, an element of degree 2
  7. a_3_9 → 0, an element of degree 3
  8. a_5_15 → 0, an element of degree 5
  9. b_5_17 → 0, an element of degree 5
  10. a_6_21 → 0, an element of degree 6
  11. c_8_37 → c_1_18, an element of degree 8

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. a_1_2 → 0, an element of degree 1
  4. a_2_3 → 0, an element of degree 2
  5. b_2_4 → c_1_22, an element of degree 2
  6. c_2_5 → c_1_02, an element of degree 2
  7. a_3_9 → 0, an element of degree 3
  8. a_5_15 → 0, an element of degree 5
  9. b_5_17 → c_1_25, an element of degree 5
  10. a_6_21 → 0, an element of degree 6
  11. c_8_37 → c_1_28 + c_1_14·c_1_24 + c_1_18, an element of degree 8

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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