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Mod-3-Cohomology of Normalizer(J2,Centre(SylowSubgroup(J2,3))), a group of order 2160

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• Normalizer(J2,Centre(SylowSubgroup(J2,3))) is a group of order 2160.
• The group order factors as 2⁴ · 3³ · 5.
• The group is defined by Group([(3,32,42)(4,86,40)(5,17,51)(6,19,21)(7,65,33)(8,62,72)(9,43,73)(10,59,74)(11,27,66)(12,81,52)(14,18,48)(16,98,96)(20,70,31)(22,56,69)(23,44,67)(24,77,91)(26,90,58)(28,83,46)(29,30,94)(34,53,37)(35,89,76)(36,75,57)(38,64,50)(39,85,84)(45,55,60)(47,49,92)(54,88,80)(61,93,79)(63,78,87)(97,100,99),(2,82,71,15)(4,98,48,87)(5,53)(6,55)(7,44,26,77)(8,22,66,12)(9,30,46,70)(10,36,97,38)(11,81,62,56)(13,41,25,95)(14,63,86,96)(16,18,78,40)(17,37)(19,60)(20,73,29,83)(21,45)(23,58,24,33)(27,52,72,69)(28,31,43,94)(34,51)(39,61,49,54)(47,80,84,79)(50,74,57,99)(59,75,100,64)(65,67,90,91)(85,93,92,88),(1,13,95)(4,8,57)(5,53,89)(6,23,58)(7,60,77)(9,79,83)(10,56,11)(12,98,50)(14,63,97)(15,82,71)(16,64,52)(17,37,76)(18,78,100)(19,44,26)(20,31,70)(21,67,90)(22,66,74)(24,33,55)(25,41,68)(27,59,69)(28,73,93)(29,47,85)(30,49,84)(34,35,51)(36,86,62)(38,81,96)(39,94,92)(40,72,75)(43,61,46)(45,91,65)(48,87,99)(54,88,80),(2,15,68,41,82,95,13,71)(3,4,83,14,65,10,94,57)(5,84,69,80,11,45,12,19)(6,17,85,22,88,27,60,81)(7,59,30,36,42,86,28,18)(8,92,34,79)(9,35,67,64,24,87,31,97)(16,90)(20,99,43,76,23,38,77,78)(21,51,39,56,54,66,55,52)(26,98)(29,75,32,40,46,48,33,74)(37,61,72,47)(44,50,91,63,70,100,73,89)(49,53,93,62)(58,96)]).
• It is non-abelian.
• It has 3-Rank 2.
• The centre of a Sylow 3-subgroup has rank 1.
• Its Sylow 3-subgroup has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 2 and depth 2.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  1  −  2·t  +  2·t2  −  t3  +  2·t4  −  3·t5  +  2·t6  −  t7  +  2·t8  −  2·t9  +  t10

  ( − 1  +  t)2 · (1  −  t  +  t2) · (1  +  t2) · (1  +  t  +  t2) · (1  −  t2  +  t4)

• The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
• The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 8 minimal generators of maximal degree 12:

1. a_3_0, a nilpotent element of degree 3
2. a_4_1, a nilpotent element of degree 4
3. b_4_0, an element of degree 4
4. a_5_0, a nilpotent element of degree 5
5. a_9_1, a nilpotent element of degree 9
6. b_10_0, an element of degree 10
7. a_11_1, a nilpotent element of degree 11
8. c_12_1, a Duflot element of degree 12

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 4 "obvious" relations:
   a_3_0², a_5_0², a_9_1², a_11_1²

Apart from that, there are 16 minimal relations of maximal degree 21:

1. a_4_1·a_3_0
2. a_4_1²
3. a_4_1·b_4_0 − a_3_0·a_5_0
4. a_4_1·a_5_0
5. a_3_0·a_9_1
6. a_4_1·a_9_1
7. b_10_0·a_3_0 − b_4_0·a_9_1
8. a_5_0·a_9_1 − a_3_0·a_11_1
9. a_4_1·b_10_0 + a_3_0·a_11_1
10. a_4_1·a_11_1
11. b_10_0·a_5_0 + b_4_0·a_11_1
12. a_5_0·a_11_1
13. b_10_0·a_9_1 + b_4_0·c_12_1·a_3_0
14. a_9_1·a_11_1 − c_12_1·a_3_0·a_5_0
15. b_10_0² + b_4_0²·c_12_1
16. b_10_0·a_11_1 − b_4_0·c_12_1·a_5_0

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 21 using the Hilbert-Poincaré criterion.
• 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_12_1, an element of degree 12
  2. b_4_0, an element of degree 4
• A Duflot regular sequence is given by c_12_1.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 14].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(E27; GF(3))

1. a_3_0 → b_2_3·a_1_1 − b_2_1·a_1_1 + b_2_0·a_1_0
2. a_4_1 → a_1_1·a_3_5 − a_1_0·a_3_4
3. b_4_0 → b_2_3² − b_2_0·b_2_2 − b_2_0·b_2_1 + b_2_0² + a_1_0·a_3_5 − a_1_0·a_3_4
4. a_5_0 → b_2_3·a_3_5 − b_2_0·a_3_4
5. a_9_1 → b_2_0³·b_2_1·a_1_1 − b_2_0⁴·a_1_1 + b_2_3·c_6_8·a_1_1 + b_2_0·c_6_8·a_1_1
      − b_2_0·c_6_8·a_1_0
6. b_10_0 → b_2_0⁴·b_2_2 − b_2_0³·a_1_1·a_3_5 − b_2_0³·a_1_0·a_3_5 − b_2_0³·a_1_0·a_3_4
      + b_2_3²·c_6_8 + b_2_0·b_2_1·c_6_8 − b_2_0²·c_6_8 + c_6_8·a_1_0·a_3_4
7. a_11_1 → b_2_0³·b_2_1·a_3_5 + b_2_0⁴·a_3_5 − b_2_0⁴·a_3_4 − b_2_0⁴·b_2_1·a_1_1
      + b_2_0⁵·a_1_1 − b_2_3·c_6_8·a_3_5 + b_2_1·c_6_8·a_3_5 − b_2_0·c_6_8·a_3_4
      − b_2_0·b_2_1·c_6_8·a_1_1 + b_2_0²·c_6_8·a_1_1
8. c_12_1 → b_2_0⁵·b_2_2 − b_2_0⁴·a_1_1·a_3_5 + b_2_0⁴·a_1_0·a_3_5 + b_2_0⁴·a_1_0·a_3_4
      − b_2_0²·b_2_2·c_6_8 + b_2_0·c_6_8·a_1_1·a_3_5 + b_2_0·c_6_8·a_1_0·a_3_5
      + b_2_0·c_6_8·a_1_0·a_3_4 − c_6_8²

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

1. a_3_0 → 0, an element of degree 3
2. a_4_1 → 0, an element of degree 4
3. b_4_0 → 0, an element of degree 4
4. a_5_0 → 0, an element of degree 5
5. a_9_1 → 0, an element of degree 9
6. b_10_0 → 0, an element of degree 10
7. a_11_1 → 0, an element of degree 11
8. c_12_1 →  − c_2_0⁶, an element of degree 12

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

1. a_3_0 → c_2_2·a_1_1, an element of degree 3
2. a_4_1 →  − c_2_2·a_1_0·a_1_1, an element of degree 4
3. b_4_0 → c_2_2², an element of degree 4
4. a_5_0 → c_2_2²·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
5. a_9_1 →  − c_2_1·c_2_2³·a_1_1 + c_2_1³·c_2_2·a_1_1, an element of degree 9
6. b_10_0 →  − c_2_1·c_2_2⁴ + c_2_1³·c_2_2², an element of degree 10
7. a_11_1 → c_2_1·c_2_2⁴·a_1_0 − c_2_1²·c_2_2³·a_1_1 − c_2_1³·c_2_2²·a_1_0
      + c_2_1⁴·c_2_2·a_1_1, an element of degree 11
8. c_12_1 →  − c_2_1²·c_2_2⁴ − c_2_1⁴·c_2_2² − c_2_1⁶, an element of degree 12

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

1. a_3_0 → c_2_2·a_1_1, an element of degree 3
2. a_4_1 → c_2_2·a_1_0·a_1_1, an element of degree 4
3. b_4_0 → c_2_2², an element of degree 4
4. a_5_0 →  − c_2_2²·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
5. a_9_1 → c_2_1·c_2_2³·a_1_1 − c_2_1³·c_2_2·a_1_1, an element of degree 9
6. b_10_0 → c_2_1·c_2_2⁴ − c_2_1³·c_2_2², an element of degree 10
7. a_11_1 → c_2_1·c_2_2⁴·a_1_0 − c_2_1²·c_2_2³·a_1_1 − c_2_1³·c_2_2²·a_1_0
      + c_2_1⁴·c_2_2·a_1_1, an element of degree 11
8. c_12_1 →  − c_2_1²·c_2_2⁴ − c_2_1⁴·c_2_2² − c_2_1⁶, an element of degree 12

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

1. a_3_0 → c_2_2·a_1_1, an element of degree 3
2. a_4_1 → c_2_2·a_1_0·a_1_1, an element of degree 4
3. b_4_0 → c_2_2², an element of degree 4
4. a_5_0 →  − c_2_2²·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
5. a_9_1 → c_2_1·c_2_2³·a_1_1 − c_2_1³·c_2_2·a_1_1, an element of degree 9
6. b_10_0 → c_2_1·c_2_2⁴ − c_2_1³·c_2_2², an element of degree 10
7. a_11_1 → c_2_1·c_2_2⁴·a_1_0 − c_2_1²·c_2_2³·a_1_1 − c_2_1³·c_2_2²·a_1_0
      + c_2_1⁴·c_2_2·a_1_1, an element of degree 11
8. c_12_1 →  − c_2_1²·c_2_2⁴ − c_2_1⁴·c_2_2² − c_2_1⁶, an element of degree 12

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

1. a_3_0 → c_2_2·a_1_1, an element of degree 3
2. a_4_1 →  − c_2_2·a_1_0·a_1_1, an element of degree 4
3. b_4_0 → c_2_2², an element of degree 4
4. a_5_0 → c_2_2²·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
5. a_9_1 →  − c_2_1·c_2_2³·a_1_1 + c_2_1³·c_2_2·a_1_1, an element of degree 9
6. b_10_0 →  − c_2_1·c_2_2⁴ + c_2_1³·c_2_2², an element of degree 10
7. a_11_1 → c_2_1·c_2_2⁴·a_1_0 − c_2_1²·c_2_2³·a_1_1 − c_2_1³·c_2_2²·a_1_0
      + c_2_1⁴·c_2_2·a_1_1, an element of degree 11
8. c_12_1 →  − c_2_1²·c_2_2⁴ − c_2_1⁴·c_2_2² − c_2_1⁶, an element of degree 12

About the group

Ring generators

Ring relations

Completion information

Restriction maps