Cohomology of group number 1643 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 4.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t6  −  t5  −  t4  +  t3  −  1)
    (t  −  1)4 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-6,-4,-4. 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. b_1_2, an element of degree 1
  4. b_1_3, an element of degree 1
  5. c_2_8, a Duflot regular element of degree 2
  6. a_4_20, a nilpotent element of degree 4
  7. a_5_7, a nilpotent element of degree 5
  8. a_5_9, a nilpotent element of degree 5
  9. b_5_30, an element of degree 5
  10. a_8_53, a nilpotent element of degree 8
  11. c_8_77, a Duflot regular element of degree 8

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Ring relations

There are 29 minimal relations of maximal degree 16:

  1. a_1_02
  2. a_1_12 + a_1_0·a_1_1
  3. a_1_1·b_1_22 + a_1_0·b_1_32
  4. a_1_0·b_1_34 + a_1_0·b_1_22·b_1_32
  5. a_4_20·a_1_1
  6. a_4_20·a_1_0
  7. a_1_0·b_1_24·b_1_3 + a_4_20·b_1_22 + a_1_0·a_5_7
  8. a_1_1·a_5_9 + a_1_1·a_5_7
  9. a_1_1·a_5_7 + a_1_0·a_5_9
  10. a_1_1·b_5_30 + a_1_0·b_1_22·b_1_33 + a_4_20·b_1_32 + a_1_1·a_5_7
  11. a_1_0·b_5_30
  12. b_1_32·a_5_7 + b_1_22·a_5_9 + a_1_0·b_1_23·b_1_33 + a_4_20·b_1_2·b_1_32
       + a_4_20·b_1_22·b_1_3 + a_1_0·b_1_3·a_5_7
  13. a_4_202
  14. a_1_0·b_1_22·b_1_3·a_5_7 + a_4_20·a_5_7
  15. a_1_0·b_1_22·b_1_3·a_5_9 + a_4_20·a_5_9
  16. a_4_20·b_5_30 + c_2_8·a_1_1·b_1_36 + c_2_8·a_4_20·b_1_22·b_1_3
       + c_2_8·a_1_0·b_1_3·a_5_7
  17. a_8_53·a_1_1 + c_2_83·a_1_0·a_1_1·b_1_3
  18. a_8_53·a_1_0 + c_2_83·a_1_0·a_1_1·b_1_3
  19. a_4_20·b_1_25·b_1_3 + a_4_20·b_1_26 + a_5_72 + a_4_20·b_1_2·a_5_7
  20. a_5_92 + a_5_7·a_5_9 + a_4_20·b_1_3·a_5_7
  21. b_5_302 + c_2_8·b_1_38 + c_2_8·b_1_24·b_1_34
  22. a_4_20·b_1_24·b_1_32 + a_4_20·b_1_26 + a_5_7·a_5_9 + a_5_72 + a_4_20·b_1_3·a_5_7
       + c_8_77·a_1_0·a_1_1
  23. a_5_7·b_5_30 + a_8_53·b_1_22 + a_4_20·b_1_3·a_5_7 + c_2_8·a_1_1·b_1_2·b_1_36
       + c_2_8·a_4_20·b_1_22·b_1_32 + c_2_8·a_4_20·b_1_23·b_1_3
       + c_2_8·a_1_0·b_1_2·b_1_3·a_5_7 + c_2_83·a_1_0·b_1_33
  24. a_5_9·b_5_30 + a_8_53·b_1_32 + a_4_20·b_1_2·b_1_35 + a_4_20·b_1_24·b_1_32
       + a_5_7·a_5_9 + a_4_20·b_1_2·a_5_9 + c_2_8·a_4_20·b_1_34 + c_2_82·a_1_1·b_1_35
       + c_2_82·a_1_1·b_1_2·b_1_34 + c_2_82·a_1_0·b_1_22·b_1_33
       + c_2_82·a_1_0·b_1_23·b_1_32 + c_2_83·a_1_1·b_1_33
  25. a_4_20·a_8_53
  26. a_8_53·b_5_30 + c_2_8·b_1_36·a_5_9 + c_2_8·b_1_24·b_1_32·a_5_9
       + c_2_8·a_1_1·b_1_2·b_1_39 + c_2_8·a_4_20·b_1_27 + c_2_8·b_1_2·a_5_7·a_5_9
       + c_2_8·b_1_2·a_5_72 + c_2_82·a_1_1·b_1_38 + c_2_82·a_4_20·b_1_35
       + c_2_82·a_4_20·b_1_2·b_1_34 + c_2_82·a_4_20·b_1_23·b_1_32
       + c_2_8·c_8_77·a_1_0·a_1_1·b_1_2 + c_2_82·a_4_20·a_5_9 + c_2_83·a_4_20·b_1_33
       + c_2_83·a_4_20·b_1_22·b_1_3 + c_2_83·a_1_0·b_1_3·a_5_9
       + c_2_83·a_1_0·b_1_3·a_5_7
  27. a_8_53·a_5_7 + c_2_8·a_4_20·b_1_22·a_5_9 + c_2_83·a_1_0·b_1_3·a_5_9
  28. a_8_53·a_5_9 + c_2_8·a_4_20·b_1_22·a_5_9 + c_2_83·a_1_0·b_1_3·a_5_9
  29. a_8_532

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 16.
  • 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_8, a Duflot regular element of degree 2
    2. c_8_77, a Duflot regular element of degree 8
    3. b_1_32 + b_1_2·b_1_3 + b_1_22, an element of degree 2
    4. b_1_32, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 8, 10].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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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. b_1_3 → 0, an element of degree 1
  5. c_2_8 → c_1_02, an element of degree 2
  6. a_4_20 → 0, an element of degree 4
  7. a_5_7 → 0, an element of degree 5
  8. a_5_9 → 0, an element of degree 5
  9. b_5_30 → 0, an element of degree 5
  10. a_8_53 → 0, an element of degree 8
  11. c_8_77 → c_1_18, an element of degree 8

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

  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. b_1_3 → c_1_3, an element of degree 1
  5. c_2_8 → c_1_02, an element of degree 2
  6. a_4_20 → 0, an element of degree 4
  7. a_5_7 → 0, an element of degree 5
  8. a_5_9 → 0, an element of degree 5
  9. b_5_30 → c_1_0·c_1_34 + c_1_0·c_1_22·c_1_32, an element of degree 5
  10. a_8_53 → 0, an element of degree 8
  11. c_8_77 → c_1_22·c_1_36 + c_1_25·c_1_33 + c_1_12·c_1_22·c_1_34
       + c_1_12·c_1_24·c_1_32 + c_1_14·c_1_34 + c_1_14·c_1_22·c_1_32
       + c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_2·c_1_36 + c_1_0·c_1_23·c_1_34
       + c_1_02·c_1_25·c_1_3 + c_1_04·c_1_34, 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