Cohomology of group number 1264 of order 128

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General information on the group

  • The group has 4 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 3.
  • 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 3.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 2) · (t8  +  1/2·t7  −  1/2·t6  +  3/2·t5  −  3/2·t4  −  t3  −  1/2·t2  −  t  −  1/2)
    (t  +  1)2 · (t  −  1)4 · (t2  +  1)3
  • The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 12 minimal generators of maximal degree 4:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_1_3, a nilpotent element of degree 1
  4. b_1_2, an element of degree 1
  5. a_3_3, a nilpotent element of degree 3
  6. a_3_5, a nilpotent element of degree 3
  7. b_3_7, an element of degree 3
  8. b_3_8, an element of degree 3
  9. b_3_9, an element of degree 3
  10. c_4_15, a Duflot regular element of degree 4
  11. c_4_16, a Duflot regular element of degree 4
  12. c_4_17, a Duflot regular element of degree 4

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Ring relations

There are 24 minimal relations of maximal degree 7:

  1. a_1_1·b_1_2 + a_1_12 + a_1_02
  2. a_1_0·b_1_2 + a_1_32 + a_1_02
  3. a_1_3·b_1_2 + a_1_1·a_1_3 + a_1_12 + a_1_0·a_1_3 + a_1_0·a_1_1 + a_1_02
  4. a_1_0·a_1_12 + a_1_02·a_1_1
  5. a_1_12·a_1_3 + a_1_0·a_1_1·a_1_3 + a_1_02·a_1_3
  6. a_1_0·a_1_32 + a_1_03
  7. b_1_2·a_3_5 + b_1_2·a_3_3 + a_1_3·b_3_7 + a_1_0·a_3_5 + a_1_0·a_3_3
  8. b_1_2·a_3_3 + a_1_1·b_3_8 + a_1_1·b_3_7 + a_1_1·a_3_5 + a_1_1·a_3_3 + a_1_0·a_3_5
       + a_1_0·a_3_3
  9. b_1_2·a_3_3 + a_1_1·b_3_7 + a_1_0·b_3_8 + a_1_0·b_3_7 + a_1_3·a_3_3 + a_1_1·a_3_3
       + a_1_0·a_3_3
  10. a_1_1·b_3_9 + a_1_1·b_3_7 + a_1_0·a_3_5
  11. b_1_2·a_3_5 + a_1_0·b_3_9 + a_1_0·b_3_7 + a_1_1·a_3_5
  12. b_1_2·a_3_3 + a_1_3·b_3_9 + a_1_3·b_3_8 + a_1_3·a_3_5 + a_1_0·a_3_3
  13. a_1_32·a_3_3 + a_1_1·a_1_3·a_3_3 + a_1_12·a_3_3 + a_1_0·a_1_1·a_3_5 + a_1_02·a_3_5
       + a_1_02·a_3_3
  14. a_1_02·b_3_7 + a_1_32·a_3_5 + a_1_12·a_3_3 + a_1_0·a_1_3·a_3_3 + a_1_0·a_1_1·a_3_5
       + a_1_0·a_1_1·a_3_3 + a_1_02·a_3_3
  15. a_1_02·b_3_8 + a_1_32·a_3_3 + a_1_12·a_3_3 + a_1_0·a_1_3·a_3_3 + a_1_02·a_3_3
  16. b_3_92 + b_3_72 + b_1_23·b_3_8 + b_1_26 + a_1_03·a_3_3 + c_4_16·b_1_22
       + c_4_15·b_1_22 + c_4_16·a_1_12 + c_4_15·a_1_12 + c_4_15·a_1_02
  17. b_3_92 + b_3_82 + b_3_72 + b_1_23·b_3_9 + a_3_5·b_3_9 + a_3_5·b_3_7 + a_1_03·a_3_5
       + c_4_15·b_1_22 + c_4_16·a_1_0·a_1_1 + c_4_15·a_1_02
  18. a_3_52 + a_1_03·a_3_5 + c_4_16·a_1_02 + c_4_15·a_1_12 + c_4_15·a_1_02
  19. a_3_5·b_3_8 + a_3_5·b_3_7 + a_3_3·b_3_9 + a_3_3·b_3_8 + a_3_3·b_3_7 + a_3_3·a_3_5
       + a_1_03·a_3_5 + a_1_03·a_3_3 + c_4_16·a_1_1·a_1_3 + c_4_15·a_1_1·a_1_3
       + c_4_15·a_1_12 + c_4_15·a_1_0·a_1_1
  20. b_3_92 + b_3_82 + b_3_72 + b_1_23·b_3_9 + a_3_52 + c_4_15·b_1_22
       + c_4_16·a_1_32 + c_4_15·a_1_32 + c_4_15·a_1_12 + c_4_15·a_1_02
  21. b_3_82 + b_3_72 + b_1_23·b_3_9 + b_1_23·b_3_8 + b_1_23·b_3_7 + b_1_26
       + a_1_03·a_3_5 + c_4_17·b_1_22 + c_4_17·a_1_02 + c_4_15·a_1_12 + c_4_15·a_1_02
  22. b_3_92 + b_1_23·b_3_8 + b_1_23·b_3_7 + b_1_26 + a_3_5·b_3_9 + a_3_5·b_3_7
       + a_3_3·b_3_9 + a_3_3·b_3_8 + a_3_52 + a_3_3·a_3_5 + a_1_03·a_3_3 + c_4_17·b_1_22
       + c_4_15·b_1_22 + c_4_17·a_1_1·a_1_3 + c_4_17·a_1_12 + c_4_17·a_1_0·a_1_1
       + c_4_15·a_1_0·a_1_1
  23. b_3_92 + b_3_72 + b_1_23·b_3_8 + b_1_26 + a_3_32 + a_1_03·a_3_5 + c_4_16·b_1_22
       + c_4_15·b_1_22 + c_4_17·a_1_32 + c_4_15·a_1_02
  24. a_1_3·a_3_3·b_3_8 + a_1_0·a_3_5·b_3_7 + a_1_0·a_3_3·b_3_8 + a_1_0·a_3_3·b_3_7
       + a_1_3·a_3_3·a_3_5 + a_1_1·a_3_3·a_3_5 + c_4_17·a_1_0·a_1_1·a_1_3 + c_4_17·a_1_03
       + c_4_15·a_1_02·a_1_1 + c_4_15·a_1_03

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 10.
  • However, the last relation was already found in degree 7 and the last generator in degree 4.
  • The following is a filter regular homogeneous system of parameters:
    1. c_4_15, a Duflot regular element of degree 4
    2. c_4_16, a Duflot regular element of degree 4
    3. c_4_17, a Duflot regular element of degree 4
    4. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 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 3

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_3 → 0, an element of degree 1
  4. b_1_2 → 0, an element of degree 1
  5. a_3_3 → 0, an element of degree 3
  6. a_3_5 → 0, an element of degree 3
  7. b_3_7 → 0, an element of degree 3
  8. b_3_8 → 0, an element of degree 3
  9. b_3_9 → 0, an element of degree 3
  10. c_4_15 → c_1_14 + c_1_04, an element of degree 4
  11. c_4_16 → c_1_24 + c_1_14, an element of degree 4
  12. c_4_17 → c_1_24, an element of degree 4

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. a_1_3 → 0, an element of degree 1
  4. b_1_2 → c_1_3, an element of degree 1
  5. a_3_3 → 0, an element of degree 3
  6. a_3_5 → 0, an element of degree 3
  7. b_3_7 → c_1_33 + c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
  8. b_3_8 → c_1_33 + c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
  9. b_3_9 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32
       + c_1_02·c_1_3, an element of degree 3
  10. c_4_15 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_1·c_1_33 + c_1_14 + c_1_0·c_1_33 + c_1_04, an element of degree 4
  11. c_4_16 → c_1_34 + c_1_22·c_1_32 + c_1_24 + c_1_12·c_1_32 + c_1_14 + c_1_0·c_1_33
       + c_1_02·c_1_32, an element of degree 4
  12. c_4_17 → c_1_34 + c_1_22·c_1_32 + c_1_24 + c_1_1·c_1_33 + c_1_12·c_1_32
       + c_1_0·c_1_33 + c_1_02·c_1_32, 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