Cohomology of group number 2245 of order 128

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General information on the group

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

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    t4  −  t3  +  t2  +  t  +  1
    (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-∞,-5,-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 4:

  1. b_1_0, an element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. b_1_3, an element of degree 1
  5. b_1_4, an element of degree 1
  6. b_4_31, an element of degree 4
  7. b_4_32, an element of degree 4
  8. c_4_33, a Duflot regular element of degree 4
  9. c_4_34, a Duflot regular element of degree 4

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Ring relations

There are 10 minimal relations of maximal degree 8:

  1. b_1_42 + b_1_0·b_1_2
  2. b_1_1·b_1_3 + b_1_0·b_1_4 + b_1_0·b_1_1
  3. b_1_0·b_1_22 + b_1_02·b_1_2
  4. b_1_0·b_1_3·b_1_4 + b_1_0·b_1_22 + b_1_0·b_1_1·b_1_4
  5. b_1_03·b_1_1·b_1_2 + b_4_31·b_1_4
  6. b_1_02·b_1_12·b_1_2 + b_1_04·b_1_2 + b_4_32·b_1_4 + b_4_31·b_1_2
  7. b_1_03·b_1_2·b_1_3 + b_1_03·b_1_1·b_1_2 + b_1_04·b_1_2 + b_4_32·b_1_0 + b_4_31·b_1_0
  8. b_4_31·b_1_02·b_1_1·b_1_4 + b_4_312 + c_4_34·b_1_02·b_1_32
       + c_4_34·b_1_02·b_1_12 + c_4_34·b_1_03·b_1_2 + c_4_33·b_1_02·b_1_32
       + c_4_33·b_1_02·b_1_12 + c_4_33·b_1_03·b_1_2
  9. b_4_31·b_1_24 + b_4_31·b_1_02·b_1_1·b_1_4 + b_4_31·b_1_03·b_1_4 + b_4_31·b_4_32
       + c_4_34·b_1_02·b_1_32 + c_4_34·b_1_02·b_1_12 + c_4_34·b_1_03·b_1_2
       + c_4_33·b_1_23·b_1_4 + c_4_33·b_1_02·b_1_32 + c_4_33·b_1_02·b_1_2·b_1_3
       + c_4_33·b_1_02·b_1_1·b_1_2 + c_4_33·b_1_02·b_1_12 + c_4_33·b_1_03·b_1_2
  10. b_4_32·b_1_24 + b_4_322 + b_4_31·b_1_23·b_1_3 + b_4_31·b_1_1·b_1_23
       + b_4_31·b_1_02·b_1_1·b_1_2 + b_4_31·b_1_03·b_1_4 + b_4_31·b_1_03·b_1_2
       + c_4_34·b_1_0·b_1_2·b_1_32 + c_4_34·b_1_0·b_1_12·b_1_2 + c_4_34·b_1_02·b_1_32
       + c_4_34·b_1_02·b_1_12 + c_4_33·b_1_24 + c_4_33·b_1_0·b_1_2·b_1_32
       + c_4_33·b_1_0·b_1_12·b_1_2 + c_4_33·b_1_02·b_1_32 + c_4_33·b_1_02·b_1_12
       + c_4_33·b_1_03·b_1_2

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 8.
  • 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_33, a Duflot regular element of degree 4
    2. c_4_34, a Duflot regular element of degree 4
    3. b_1_32 + b_1_2·b_1_4 + b_1_2·b_1_3 + b_1_22 + b_1_1·b_1_2 + b_1_12 + b_1_0·b_1_4
         + b_1_0·b_1_3 + b_1_0·b_1_1 + b_1_02, an element of degree 2
    4. b_1_2·b_1_32 + b_1_22·b_1_3 + b_1_1·b_1_22 + b_1_12·b_1_2 + b_1_0·b_1_32
         + b_1_0·b_1_1·b_1_4 + b_1_0·b_1_12 + b_1_02·b_1_4 + b_1_02·b_1_3 + b_1_02·b_1_1, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 9].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
  • We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 2 elements of degree 2.

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. b_1_4 → 0, an element of degree 1
  6. b_4_31 → 0, an element of degree 4
  7. b_4_32 → 0, an element of degree 4
  8. c_4_33 → c_1_04, an element of degree 4
  9. c_4_34 → c_1_14 + c_1_04, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 3

  1. b_1_0 → c_1_2, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. b_1_4 → c_1_2, an element of degree 1
  6. b_4_31 → c_1_24, an element of degree 4
  7. b_4_32 → c_1_24, an element of degree 4
  8. c_4_33 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  9. c_4_34 → c_1_24 + c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4

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

  1. b_1_0 → c_1_2, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. b_1_4 → 0, an element of degree 1
  6. b_4_31 → c_1_1·c_1_22·c_1_3 + c_1_12·c_1_2·c_1_3, an element of degree 4
  7. b_4_32 → c_1_1·c_1_22·c_1_3 + c_1_12·c_1_2·c_1_3, an element of degree 4
  8. c_4_33 → c_1_1·c_1_22·c_1_3 + c_1_12·c_1_2·c_1_3 + 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_34 → c_1_1·c_1_22·c_1_3 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + 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

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

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. b_1_4 → 0, an element of degree 1
  6. b_4_31 → 0, an element of degree 4
  7. b_4_32 → c_1_0·c_1_2·c_1_32 + c_1_02·c_1_32, an element of degree 4
  8. c_4_33 → c_1_0·c_1_2·c_1_32 + c_1_02·c_1_32 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  9. c_4_34 → c_1_12·c_1_32 + c_1_14 + c_1_0·c_1_2·c_1_32 + c_1_02·c_1_32 + c_1_02·c_1_22
       + c_1_04, an element of degree 4

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

  1. b_1_0 → c_1_3, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. b_1_4 → 0, an element of degree 1
  6. b_4_31 → c_1_1·c_1_33 + c_1_1·c_1_2·c_1_32 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3, an element of degree 4
  7. b_4_32 → c_1_1·c_1_33 + c_1_1·c_1_2·c_1_32 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3, an element of degree 4
  8. c_4_33 → c_1_1·c_1_33 + c_1_1·c_1_2·c_1_32 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3
       + 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_34 → c_1_1·c_1_33 + c_1_1·c_1_2·c_1_32 + c_1_12·c_1_2·c_1_3 + 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

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

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. b_1_4 → 0, an element of degree 1
  6. b_4_31 → 0, an element of degree 4
  7. b_4_32 → c_1_0·c_1_22·c_1_3 + c_1_02·c_1_22, an element of degree 4
  8. c_4_33 → c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_22 + c_1_04, an element of degree 4
  9. c_4_34 → c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_22
       + c_1_04, 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