Cohomology of group number 2234 of order 128

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

General information on the group

  • The group has 5 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    (t2  +  t  +  1) · (t3  −  t  −  1)
    (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 8 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. a_1_2, a nilpotent element of degree 1
  4. a_1_3, a nilpotent element of degree 1
  5. b_1_4, an element of degree 1
  6. c_4_31, a Duflot regular element of degree 4
  7. b_6_44, an element of degree 6
  8. c_8_57, a Duflot regular element of degree 8

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Ring relations

There are 7 minimal relations of maximal degree 12:

  1. a_1_22 + a_1_0·a_1_2 + a_1_02
  2. a_1_0·b_1_4 + a_1_32 + a_1_1·a_1_3 + a_1_12 + a_1_0·a_1_1
  3. a_1_03
  4. a_1_32·b_1_4 + a_1_1·a_1_3·b_1_4 + a_1_12·b_1_4 + a_1_13 + a_1_0·a_1_32
       + a_1_0·a_1_1·a_1_3 + a_1_0·a_1_12
  5. a_1_13·b_1_42 + a_1_14·b_1_4 + a_1_13·a_1_32 + a_1_14·a_1_3 + a_1_0·a_1_14
  6. b_6_44·a_1_0 + a_1_13·a_1_2·a_1_33 + a_1_15·a_1_32 + a_1_0·a_1_14·a_1_2·a_1_3
       + a_1_0·a_1_15·a_1_3 + a_1_0·a_1_15·a_1_2 + c_4_31·a_1_13 + c_4_31·a_1_0·a_1_32
       + c_4_31·a_1_0·a_1_1·a_1_3 + c_4_31·a_1_0·a_1_12
  7. b_6_442 + c_4_31·b_1_48 + c_4_312·b_1_44 + c_4_312·a_1_12·b_1_42

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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_31, a Duflot regular element of degree 4
    2. c_8_57, a Duflot regular element of degree 8
    3. b_1_42, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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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. a_1_2 → 0, an element of degree 1
  4. a_1_3 → 0, an element of degree 1
  5. b_1_4 → 0, an element of degree 1
  6. c_4_31 → c_1_14, an element of degree 4
  7. b_6_44 → 0, an element of degree 6
  8. c_8_57 → c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. 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_2 → 0, an element of degree 1
  4. a_1_3 → 0, an element of degree 1
  5. b_1_4 → c_1_2, an element of degree 1
  6. c_4_31 → c_1_14, an element of degree 4
  7. b_6_44 → c_1_12·c_1_24 + c_1_14·c_1_22, an element of degree 6
  8. c_8_57 → c_1_04·c_1_24 + c_1_08, 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