Cohomology of group number 1126 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 4.
  • 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 4.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t4  +  2·t3  +  t2  +  2·t  +  1
    (t  +  1)2 · (t  −  1)4 · (t2  +  1)
  • The a-invariants are -∞,-∞,-∞,-∞,-4. 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_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. c_1_3, a Duflot regular element of degree 1
  5. c_2_7, a Duflot regular element of degree 2
  6. c_2_8, a Duflot regular element of degree 2
  7. a_3_15, a nilpotent element of degree 3
  8. a_3_16, a nilpotent element of degree 3
  9. a_3_17, a nilpotent element of degree 3
  10. c_4_30, a Duflot regular element of degree 4

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Ring relations

There are 16 minimal relations of maximal degree 6:

  1. a_1_1·a_1_2
  2. a_1_12 + a_1_0·a_1_2
  3. a_1_22 + a_1_12 + a_1_0·a_1_1 + a_1_02
  4. a_1_02·a_1_1
  5. a_1_2·a_3_15 + a_1_1·a_3_16 + a_1_1·a_3_15 + c_2_8·a_1_0·a_1_2 + c_2_8·a_1_0·a_1_1
       + c_2_8·a_1_02 + c_2_7·a_1_0·a_1_2 + c_2_7·a_1_0·a_1_1
  6. a_1_1·a_3_15 + a_1_0·a_3_16 + a_1_0·a_3_15 + c_2_7·a_1_0·a_1_1 + c_2_7·a_1_02
  7. a_1_2·a_3_16 + a_1_2·a_3_15 + c_2_7·a_1_0·a_1_2
  8. a_1_1·a_3_17
  9. a_1_2·a_3_15 + a_1_0·a_3_17 + c_2_8·a_1_0·a_1_2 + c_2_8·a_1_0·a_1_1 + c_2_8·a_1_02
  10. a_1_2·a_3_17 + a_1_2·a_3_15 + a_1_1·a_3_15 + a_1_0·a_3_15 + c_2_8·a_1_0·a_1_1
       + c_2_8·a_1_02
  11. a_3_172 + a_3_162 + a_3_15·a_3_16 + a_3_152 + c_2_7·a_1_0·a_3_16
       + c_2_82·a_1_0·a_1_2 + c_2_82·a_1_0·a_1_1 + c_2_82·a_1_02 + c_2_72·a_1_0·a_1_2
       + c_2_72·a_1_0·a_1_1
  12. a_3_162 + a_3_15·a_3_17 + a_3_152 + c_2_8·a_1_0·a_3_17 + c_2_8·a_1_0·a_3_16
       + c_2_82·a_1_0·a_1_2 + c_2_7·c_2_8·a_1_0·a_1_1 + c_2_7·c_2_8·a_1_02
       + c_2_72·a_1_0·a_1_2 + c_2_72·a_1_02
  13. a_3_16·a_3_17 + a_3_162 + a_3_152 + c_2_8·a_1_0·a_3_17 + c_2_8·a_1_0·a_3_16
       + c_2_7·a_1_0·a_3_17 + c_2_82·a_1_0·a_1_2 + c_2_7·c_2_8·a_1_0·a_1_1
       + c_2_7·c_2_8·a_1_02 + c_2_72·a_1_0·a_1_2 + c_2_72·a_1_02
  14. a_3_15·a_3_16 + a_3_152 + c_4_30·a_1_0·a_1_1 + c_2_7·a_1_0·a_3_16
       + c_2_72·a_1_0·a_1_1 + c_2_72·a_1_02
  15. a_3_152 + c_4_30·a_1_02 + c_2_82·a_1_0·a_1_2 + c_2_82·a_1_0·a_1_1
       + c_2_82·a_1_02
  16. a_3_162 + a_3_152 + c_4_30·a_1_0·a_1_2 + c_2_72·a_1_0·a_1_2 + c_2_72·a_1_02

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 6.
  • 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_1_3, a Duflot regular element of degree 1
    2. c_2_7, a Duflot regular element of degree 2
    3. c_2_8, a Duflot regular element of degree 2
    4. c_4_30, a Duflot regular element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 5].
  • 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 4

  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. c_1_3 → c_1_0, an element of degree 1
  5. c_2_7 → c_1_32, an element of degree 2
  6. c_2_8 → c_1_12, an element of degree 2
  7. a_3_15 → 0, an element of degree 3
  8. a_3_16 → 0, an element of degree 3
  9. a_3_17 → 0, an element of degree 3
  10. c_4_30 → c_1_24, 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