Cohomology of group number 164 of order 128

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General information on the group

  • The group has 3 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
    t2  +  t  +  1
    (t  +  1)2 · (t  −  1)4
  • 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 9 minimal generators of maximal degree 2:

  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_2, a nilpotent element of degree 2
  5. a_2_3, a nilpotent element of degree 2
  6. c_2_4, a Duflot regular element of degree 2
  7. c_2_5, a Duflot regular element of degree 2
  8. c_2_6, a Duflot regular element of degree 2
  9. c_2_7, a Duflot regular element of degree 2

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Ring relations

There are 10 minimal relations of maximal degree 4:

  1. a_1_02
  2. a_1_12
  3. a_1_0·a_1_1
  4. a_1_22
  5. a_2_2·a_1_0
  6. a_2_3·a_1_1
  7. a_2_3·a_1_0 + a_2_2·a_1_1
  8. a_2_22
  9. a_2_2·a_2_3
  10. a_2_32

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 4.
  • 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_4, a Duflot regular element of degree 2
    2. c_2_5, a Duflot regular element of degree 2
    3. c_2_6, a Duflot regular element of degree 2
    4. c_2_7, a Duflot regular element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 4].
  • 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. a_2_2 → 0, an element of degree 2
  5. a_2_3 → 0, an element of degree 2
  6. c_2_4 → c_1_12, an element of degree 2
  7. c_2_5 → c_1_02, an element of degree 2
  8. c_2_6 → c_1_22, an element of degree 2
  9. c_2_7 → c_1_32, an element of degree 2

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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