Cohomology of group number 4 of order 343

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 343


General information on the group

  • The group is also known as M343, the Extraspecial 7-group of order 343 and exponent 49.
  • The group has 2 minimal generators and exponent 49.
  • It is non-abelian.
  • It has p-Rank 2.
  • Its center has rank 1.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.


Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 2 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    1

    (t  −  1)2 · (t6  −  t5  +  t4  −  t3  +  t2  −  t  +  1) · (t6  +  t5  +  t4  +  t3  +  t2  +  t  +  1)
  • The a-invariants are -∞,-2,-2. 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 14:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. b_2_1, an element of degree 2
  4. a_3_1, a nilpotent element of degree 3
  5. a_5_1, a nilpotent element of degree 5
  6. a_7_1, a nilpotent element of degree 7
  7. a_9_1, a nilpotent element of degree 9
  8. a_11_1, a nilpotent element of degree 11
  9. a_13_1, a nilpotent element of degree 13
  10. c_14_2, a Duflot regular element of degree 14

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Ring relations

There are 8 "obvious" relations:
   a_1_02, a_1_12, a_3_12, a_5_12, a_7_12, a_9_12, a_11_12, a_13_12

Apart from that, there are 27 minimal relations of maximal degree 24:

  1. b_2_1·a_1_0
  2. a_1_0·a_3_1
  3. b_2_1·a_3_1
  4. a_1_0·a_5_1
  5. b_2_1·a_5_1
  6. a_3_1·a_5_1
  7. a_1_0·a_7_1
  8. b_2_1·a_7_1
  9. a_3_1·a_7_1
  10. a_1_0·a_9_1
  11. b_2_1·a_9_1
  12. a_5_1·a_7_1
  13. a_3_1·a_9_1
  14. a_1_0·a_11_1
  15. b_2_1·a_11_1
  16. a_5_1·a_9_1
  17. a_3_1·a_11_1
  18. a_1_0·a_13_1
  19. a_7_1·a_9_1
  20. a_5_1·a_11_1
  21. a_3_1·a_13_1
  22. a_7_1·a_11_1
  23. a_5_1·a_13_1
  24. a_9_1·a_11_1
  25. a_7_1·a_13_1
  26. a_9_1·a_13_1
  27. a_11_1·a_13_1


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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 24.
  • 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_14_2, a Duflot regular element of degree 14
    2. b_2_16, an element of degree 12
  • The Raw Filter Degree Type of that HSOP is [-1, 12, 24].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].
  • We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 1 elements of degree 2.


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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  1. a_1_00, an element of degree 1
  2. a_1_10, an element of degree 1
  3. b_2_10, an element of degree 2
  4. a_3_10, an element of degree 3
  5. a_5_10, an element of degree 5
  6. a_7_10, an element of degree 7
  7. a_9_10, an element of degree 9
  8. a_11_10, an element of degree 11
  9. a_13_10, an element of degree 13
  10. c_14_2 − c_2_07, an element of degree 14

Restriction map to a maximal el. ab. subgp. of rank 2

  1. a_1_00, an element of degree 1
  2. a_1_1a_1_1, an element of degree 1
  3. b_2_1c_2_2, an element of degree 2
  4. a_3_10, an element of degree 3
  5. a_5_10, an element of degree 5
  6. a_7_10, an element of degree 7
  7. a_9_10, an element of degree 9
  8. a_11_10, an element of degree 11
  9. a_13_1c_2_26·a_1_0 − c_2_1·c_2_25·a_1_1, an element of degree 13
  10. c_14_2 − 2·c_2_26·a_1_0·a_1_1 + c_2_1·c_2_26 − c_2_17, an element of degree 14


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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