David J. Green

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Cohomology of group number 47 of order 243

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General information on the group

• The group has 3 minimal generators and exponent 9.
• 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

  t8  −  t7  +  2·t6  −  2·t5  +  t4  −  t3  −  1

  (t  −  1)3 · (t2  −  t  +  1)2 · (t2  +  t  +  1)2

• 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 14 minimal generators of maximal degree 7:

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. b_2_3, an element of degree 2
5. a_3_2, a nilpotent element of degree 3
6. a_3_3, a nilpotent element of degree 3
7. a_3_4, a nilpotent element of degree 3
8. a_5_3, a nilpotent element of degree 5
9. a_5_4, a nilpotent element of degree 5
10. a_5_5, a nilpotent element of degree 5
11. a_5_6, a nilpotent element of degree 5
12. c_6_8, a Duflot regular element of degree 6
13. c_6_9, a Duflot regular element of degree 6
14. a_7_12, a nilpotent element of degree 7

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Ring relations

There are 11 "obvious" relations:
   a_1_0², a_1_1², a_1_2², a_3_2², a_3_3², a_3_4², a_5_3², a_5_4², a_5_5², a_5_6², a_7_12²

Apart from that, there are 44 minimal relations of maximal degree 12:

1. b_2_3·a_1_1
2. b_2_3·a_1_2
3. a_1_1·a_3_2
4. a_1_2·a_3_2 + a_1_1·a_3_3
5. a_1_2·a_3_3 + a_1_1·a_3_4
6. a_1_2·a_3_4
7. b_2_3·a_3_2
8. b_2_3·a_3_3
9. b_2_3·a_3_4
10. a_1_1·a_5_3
11.  − a_3_2·a_3_3 + a_1_2·a_5_3
12. a_3_2·a_3_3 + a_1_1·a_5_4
13. a_3_2·a_3_4 + a_1_2·a_5_4
14.  − a_3_2·a_3_4 + a_3_2·a_3_3 + a_1_1·a_5_5
15.  − a_3_3·a_3_4 + a_3_2·a_3_4 + a_1_2·a_5_5
16. a_3_3·a_3_4 + a_1_1·a_5_6
17. a_1_2·a_5_6
18. b_2_3·a_5_4
19. b_2_3·a_5_5
20. a_3_2·a_5_3
21.  − a_3_4·a_5_3 + a_3_3·a_5_4
22. a_3_3·a_5_3 + a_3_2·a_5_4
23.  − a_3_4·a_5_4 − a_3_4·a_5_3 + a_3_3·a_5_5
24.  − a_3_4·a_5_3 + a_3_3·a_5_3 + a_3_2·a_5_5
25. a_3_4·a_5_5 − a_3_4·a_5_4 + a_3_3·a_5_6
26.  − a_3_4·a_5_4 + a_3_2·a_5_6
27. a_3_4·a_5_6
28.  − a_3_3·a_5_3 + c_6_8·a_1_1·a_1_2
29.  − a_3_4·a_5_5 + a_3_4·a_5_4 + c_6_9·a_1_1·a_1_2
30. a_3_4·a_5_3 + a_1_1·a_7_12
31. a_3_4·a_5_4 + a_1_2·a_7_12
32. b_2_3·a_7_12
33. a_5_4·a_5_5
34.  − a_5_3·a_5_4 + c_6_8·a_1_1·a_3_3
35. a_5_3·a_5_5 − a_5_3·a_5_4 + c_6_8·a_1_1·a_3_4
36. a_5_4·a_5_6 + c_6_9·a_1_1·a_3_3
37.  − a_5_5·a_5_6 + a_5_4·a_5_6 + c_6_9·a_1_1·a_3_4
38. a_3_3·a_7_12
39.  − a_5_3·a_5_5 + a_5_3·a_5_4 + a_3_2·a_7_12
40. a_5_4·a_5_6 + a_3_4·a_7_12
41. a_5_3·a_7_12 − c_6_8·a_1_1·a_5_5 + c_6_8·a_1_1·a_5_4
42. a_5_5·a_7_12 − c_6_9·a_1_1·a_5_4 + c_6_8·a_1_1·a_5_6
43. a_5_4·a_7_12 + c_6_8·a_1_1·a_5_6
44. a_5_6·a_7_12 + c_6_9·a_1_1·a_5_5 − c_6_9·a_1_1·a_5_4

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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_6_8, a Duflot regular element of degree 6
  2. c_6_9, a Duflot regular element of degree 6
  3. b_2_3, 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. b_2_3 → 0, an element of degree 2
5. a_3_2 → 0, an element of degree 3
6. a_3_3 → 0, an element of degree 3
7. a_3_4 → 0, an element of degree 3
8. a_5_3 → 0, an element of degree 5
9. a_5_4 → 0, an element of degree 5
10. a_5_5 → 0, an element of degree 5
11. a_5_6 → 0, an element of degree 5
12. c_6_8 →  − c_2_1³, an element of degree 6
13. c_6_9 →  − c_2_2³, an element of degree 6
14. a_7_12 → 0, an element of degree 7

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

1. a_1_0 → a_1_2, 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. b_2_3 → c_2_5, an element of degree 2
5. a_3_2 → 0, an element of degree 3
6. a_3_3 → 0, an element of degree 3
7. a_3_4 → 0, an element of degree 3
8. a_5_3 → c_2_5²·a_1_0 − c_2_3·c_2_5·a_1_2, an element of degree 5
9. a_5_4 → 0, an element of degree 5
10. a_5_5 → 0, an element of degree 5
11. a_5_6 → c_2_5²·a_1_1 − c_2_4·c_2_5·a_1_2, an element of degree 5
12. c_6_8 → c_2_5²·a_1_0·a_1_2 + c_2_3·c_2_5² − c_2_3³, an element of degree 6
13. c_6_9 → c_2_5²·a_1_1·a_1_2 + c_2_4·c_2_5² − c_2_4³, an element of degree 6
14. a_7_12 → 0, an element of degree 7

About the group

Ring generators

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Completion information

Restriction maps

Back to groups of order 243