Cohomology of group number 1928 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 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3 and 4, respectively.

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
    (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-4,-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 6:

  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_3_4, a nilpotent element of degree 3
  6. b_3_10, an element of degree 3
  7. a_4_14, a nilpotent element of degree 4
  8. c_4_16, a Duflot regular element of degree 4
  9. c_4_17, a Duflot regular element of degree 4
  10. b_5_27, an element of degree 5
  11. a_6_25, a nilpotent element of degree 6

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Ring relations

There are 29 minimal relations of maximal degree 12:

  1. a_1_2·b_1_0
  2. b_1_0·b_1_3 + b_1_0·b_1_1 + a_1_2·b_1_1
  3. b_1_0·b_1_12 + a_1_23
  4. a_1_2·b_1_1·b_1_3
  5. b_1_03·b_1_1 + b_1_0·a_3_4
  6. b_1_3·b_3_10 + b_1_1·b_3_10 + b_1_12·b_1_32 + b_1_13·b_1_3 + b_1_1·a_3_4
       + a_1_23·b_1_3
  7. a_1_2·b_3_10
  8. b_1_12·b_3_10 + b_1_14·b_1_3 + b_1_1·b_1_3·a_3_4 + a_4_14·b_1_3 + a_1_22·b_1_33
       + a_1_22·a_3_4
  9. a_4_14·b_1_0
  10. a_4_14·a_1_2
  11. b_1_02·b_1_1·b_3_10 + a_3_4·b_3_10 + b_1_12·b_1_3·a_3_4
  12. b_3_102 + b_1_14·b_1_32 + b_1_03·b_3_10 + b_1_03·a_3_4 + c_4_16·b_1_02
  13. a_3_42 + a_1_2·b_1_32·a_3_4 + a_1_22·b_1_34 + a_1_22·b_1_3·a_3_4
       + c_4_16·a_1_22
  14. b_1_0·b_5_27 + c_4_16·b_1_0·b_1_1
  15. a_1_2·b_5_27 + a_3_42 + a_1_22·b_1_34 + c_4_16·a_1_2·b_1_1
  16. a_4_14·b_3_10 + a_4_14·b_1_12·b_1_3
  17. a_4_14·a_3_4 + a_1_22·b_1_32·a_3_4
  18. b_1_0·a_3_4·b_3_10 + a_6_25·b_1_0 + c_4_16·b_1_02·b_1_1
  19. a_6_25·a_1_2 + a_1_22·b_1_32·a_3_4 + c_4_17·a_1_22·b_1_3 + c_4_16·a_1_23
  20. a_4_142
  21. b_3_10·b_5_27 + b_1_12·b_1_3·b_5_27 + b_1_14·b_1_3·a_3_4 + b_1_15·a_3_4
       + a_6_25·b_1_1·b_1_3 + a_4_14·b_1_1·b_1_33 + a_4_14·b_1_13·b_1_3
       + c_4_16·b_1_1·b_3_10 + c_4_16·b_1_13·b_1_3 + c_4_16·a_1_23·b_1_3
  22. b_3_10·b_5_27 + b_1_12·b_1_3·b_5_27 + a_3_4·b_5_27 + b_1_15·a_3_4 + a_6_25·b_1_32
       + a_4_14·b_1_34 + a_4_14·b_1_13·b_1_3 + a_1_22·b_1_33·a_3_4 + c_4_16·b_1_1·b_3_10
       + c_4_16·b_1_13·b_1_3 + c_4_17·a_1_2·b_1_33 + c_4_16·b_1_1·a_3_4
       + c_4_16·a_1_2·a_3_4
  23. a_6_25·b_1_1·b_1_32 + a_6_25·b_1_12·b_1_3 + a_6_25·b_1_13 + a_4_14·b_5_27
       + a_4_14·b_1_1·b_1_34 + a_4_14·b_1_12·b_1_33 + a_4_14·b_1_13·b_1_32
       + a_4_14·b_1_14·b_1_3 + a_1_22·b_1_34·a_3_4 + a_4_14·c_4_16·b_1_1
  24. a_6_25·b_3_10 + a_6_25·b_1_12·b_1_3 + a_6_25·b_1_03 + c_4_16·b_1_0·b_1_1·b_3_10
  25. a_6_25·a_3_4 + a_1_22·b_1_34·a_3_4 + c_4_17·a_1_2·b_1_3·a_3_4 + c_4_16·a_1_22·a_3_4
  26. b_5_272 + b_1_12·b_1_38 + b_1_13·b_1_32·b_5_27 + b_1_13·b_1_37
       + b_1_14·b_1_3·b_5_27 + b_1_14·b_1_36 + b_1_15·b_1_35 + b_1_17·b_1_33
       + b_1_18·b_1_32 + b_1_17·a_3_4 + a_4_14·b_1_1·b_1_35 + a_4_14·b_1_12·b_1_34
       + a_4_14·b_1_13·b_1_33 + a_4_14·b_1_16 + a_1_2·b_1_36·a_3_4 + a_1_22·b_1_38
       + a_1_22·b_1_35·a_3_4 + c_4_17·b_1_12·b_1_34 + c_4_17·b_1_14·b_1_32
       + c_4_16·b_1_12·b_1_34 + c_4_16·b_1_15·b_1_3 + c_4_16·b_1_16
       + c_4_16·a_1_22·b_1_34 + c_4_162·b_1_12 + c_4_162·a_1_22
  27. a_4_14·a_6_25
  28. a_6_25·b_5_27 + a_6_25·b_1_14·b_1_3 + a_6_25·b_1_15 + a_4_14·b_1_32·b_5_27
       + a_4_14·b_1_1·b_1_36 + a_4_14·b_1_12·b_5_27 + a_4_14·b_1_16·b_1_3
       + c_4_17·b_1_14·a_3_4 + c_4_16·a_6_25·b_1_1 + a_4_14·c_4_17·b_1_1·b_1_32
       + a_4_14·c_4_17·b_1_12·b_1_3 + a_4_14·c_4_16·b_1_12·b_1_3
       + c_4_17·a_1_2·b_1_33·a_3_4 + c_4_17·a_1_22·b_1_32·a_3_4
       + c_4_16·c_4_17·a_1_22·b_1_3 + c_4_162·a_1_23
  29. a_6_252 + c_4_172·a_1_22·b_1_32

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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_4_16, a Duflot regular element of degree 4
    2. c_4_17, a Duflot regular element of degree 4
    3. b_1_32 + b_1_1·b_1_3 + b_1_12 + b_1_02, an element of degree 2
    4. b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 6, 8].
  • 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_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_3_4 → 0, an element of degree 3
  6. b_3_10 → 0, an element of degree 3
  7. a_4_14 → 0, an element of degree 4
  8. c_4_16 → c_1_04, an element of degree 4
  9. c_4_17 → c_1_14 + c_1_04, an element of degree 4
  10. b_5_27 → 0, an element of degree 5
  11. a_6_25 → 0, an element of degree 6

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 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. a_3_4 → 0, an element of degree 3
  6. b_3_10 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  7. a_4_14 → 0, an element of degree 4
  8. c_4_16 → c_1_0·c_1_23 + c_1_04, an element of degree 4
  9. c_4_17 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  10. b_5_27 → 0, an element of degree 5
  11. a_6_25 → 0, an element of degree 6

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

  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_3, an element of degree 1
  4. b_1_3 → c_1_2, an element of degree 1
  5. a_3_4 → 0, an element of degree 3
  6. b_3_10 → c_1_2·c_1_32, an element of degree 3
  7. a_4_14 → 0, an element of degree 4
  8. c_4_16 → c_1_22·c_1_32 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32
       + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  9. c_4_17 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_12·c_1_32 + c_1_14
       + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3
       + c_1_02·c_1_22 + c_1_04, an element of degree 4
  10. b_5_27 → c_1_2·c_1_34 + c_1_24·c_1_3 + c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3
       + c_1_0·c_1_34 + c_1_0·c_1_2·c_1_33 + c_1_0·c_1_22·c_1_32
       + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3 + c_1_04·c_1_3, an element of degree 5
  11. a_6_25 → 0, an element of degree 6

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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