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Mod-3-Cohomology of L3(3).2, a group of order 11232

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• L3(3).2 is a group of order 11232.
• The group order factors as 2⁵ · 3³ · 13.
• The group is defined by Group([(1,2)(3,5)(4,6)(7,10)(8,9)(11,15)(12,16)(13,18)(14,20)(17,22)(19,24)(21,23)(25,26),(1,3)(2,4,7,11)(5,8,12,17)(6,9,13,19)(10,14)(15,21,25,22)(16,20)(18,23)(24,26)]).
• 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  +  3·t2  −  3·t3  +  4·t4  −  5·t5  +  6·t6  −  6·t7  +  7·t8  −  8·t9  +  10·t10  −  9·t11  +  9·t12  −  9·t13  +  10·t14  −  8·t15  +  7·t16  −  6·t17  +  6·t18  −  5·t19  +  4·t20  −  3·t21  +  3·t22  −  2·t23  +  t24

  ( − 1  +  t)2 · (1  −  t  +  t2) · (1  +  t  +  t2) · (1  +  t2)2 · (1  −  t2  +  t4) · (1  +  t4) · (1  +  t8)

• 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 16:

1. a_3_0, a nilpotent element of degree 3
2. b_4_0, an element of degree 4
3. a_10_0, a nilpotent element of degree 10
4. a_11_1, a nilpotent element of degree 11
5. a_11_0, a nilpotent element of degree 11
6. c_12_1, a Duflot element of degree 12
7. a_15_1, a nilpotent element of degree 15
8. b_16_1, an element of degree 16

About the group

Ring generators

Ring relations

Completion information

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Ring relations

There are 4 "obvious" relations:
   a_3_0², a_11_0², a_11_1², a_15_1²

Apart from that, there are 16 minimal relations of maximal degree 27:

1. a_10_0·a_3_0
2. a_3_0·a_11_0
3. b_4_0·a_10_0 − a_3_0·a_11_1
4. b_4_0·a_11_0
5. a_3_0·a_15_1 − b_4_0·a_3_0·a_11_1
6. b_4_0·a_15_1 − b_4_0²·a_11_1 + b_4_0⁴·a_3_0 + b_4_0·c_12_1·a_3_0
7. b_16_1·a_3_0 − b_4_0⁴·a_3_0 − b_4_0·c_12_1·a_3_0
8. a_10_0²
9. b_4_0·b_16_1 − b_4_0⁵ − b_4_0²·c_12_1
10. a_10_0·a_11_0
11. a_10_0·a_11_1
12. a_11_0·a_11_1
13. a_10_0·a_15_1
14. a_11_1·a_15_1 − a_11_0·a_15_1 − b_4_0³·a_3_0·a_11_1 − c_12_1·a_3_0·a_11_1
15. a_10_0·b_16_1 − a_11_0·a_15_1 − b_4_0³·a_3_0·a_11_1 − c_12_1·a_3_0·a_11_1
16. b_16_1·a_11_1 − b_16_1·a_11_0 − b_4_0⁴·a_11_1 − b_4_0·c_12_1·a_11_1

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 27 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_16_1, an element of degree 16
• A Duflot regular sequence is given by c_12_1.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 26].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(216,87); GF(3))

1. a_3_0 → a_3_1 + a_3_0
2. b_4_0 → b_4_1 + b_4_0
3. a_10_0 → a_10_0
4. a_11_1 → a_11_1 + b_4_0²·a_3_0
5. a_11_0 → a_11_3
6. c_12_1 → c_12_2
7. a_15_1 → b_4_0·a_11_1 − c_12_2·a_3_0
8. b_16_1 → b_4_0⁴ + b_4_0·c_12_2

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. b_4_0 → 0, an element of degree 4
3. a_10_0 → 0, an element of degree 10
4. a_11_1 → 0, an element of degree 11
5. a_11_0 → 0, an element of degree 11
6. c_12_1 → c_2_0⁶, an element of degree 12
7. a_15_1 → 0, an element of degree 15
8. b_16_1 → 0, an element of degree 16

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

1. a_3_0 → 0, an element of degree 3
2. b_4_0 → 0, an element of degree 4
3. a_10_0 → c_2_1·c_2_2³·a_1_0·a_1_1 − c_2_1³·c_2_2·a_1_0·a_1_1, an element of degree 10
4. 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
5. a_11_0 →  − 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
6. c_12_1 → c_2_2⁶ + c_2_1²·c_2_2⁴ + c_2_1⁴·c_2_2² + c_2_1⁶, an element of degree 12
7. a_15_1 → c_2_1·c_2_2⁶·a_1_0 − c_2_1³·c_2_2⁴·a_1_0 − c_2_1⁴·c_2_2³·a_1_1
      + c_2_1⁶·c_2_2·a_1_1, an element of degree 15
8. b_16_1 →  − c_2_1²·c_2_2⁶ − c_2_1⁴·c_2_2⁴ − c_2_1⁶·c_2_2², an element of degree 16

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. b_4_0 → c_2_2², an element of degree 4
3. a_10_0 →  − c_2_1·c_2_2³·a_1_0·a_1_1 + c_2_1³·c_2_2·a_1_0·a_1_1, an element of degree 10
4. a_11_1 → c_2_2⁵·a_1_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
5. a_11_0 → 0, an element of degree 11
6. c_12_1 → c_2_1²·c_2_2⁴ + c_2_1⁴·c_2_2² + c_2_1⁶, an element of degree 12
7. a_15_1 → c_2_1·c_2_2⁶·a_1_0 + c_2_1²·c_2_2⁵·a_1_1 − c_2_1³·c_2_2⁴·a_1_0
      − c_2_1⁶·c_2_2·a_1_1, an element of degree 15
8. b_16_1 → c_2_2⁸ + c_2_1²·c_2_2⁶ + c_2_1⁴·c_2_2⁴ + c_2_1⁶·c_2_2², an element of degree 16

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

1. a_3_0 → 0, an element of degree 3
2. b_4_0 → 0, an element of degree 4
3. a_10_0 → c_2_1·c_2_2³·a_1_0·a_1_1 − c_2_1³·c_2_2·a_1_0·a_1_1, an element of degree 10
4. 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
5. a_11_0 →  − 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
6. c_12_1 → c_2_2⁶ + c_2_1²·c_2_2⁴ + c_2_1⁴·c_2_2² + c_2_1⁶, an element of degree 12
7. a_15_1 → c_2_1·c_2_2⁶·a_1_0 − c_2_1³·c_2_2⁴·a_1_0 − c_2_1⁴·c_2_2³·a_1_1
      + c_2_1⁶·c_2_2·a_1_1, an element of degree 15
8. b_16_1 →  − c_2_1²·c_2_2⁶ − c_2_1⁴·c_2_2⁴ − c_2_1⁶·c_2_2², an element of degree 16

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. b_4_0 → c_2_2², an element of degree 4
3. a_10_0 →  − c_2_1·c_2_2³·a_1_0·a_1_1 + c_2_1³·c_2_2·a_1_0·a_1_1, an element of degree 10
4. a_11_1 → c_2_2⁵·a_1_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
5. a_11_0 → 0, an element of degree 11
6. c_12_1 → c_2_1²·c_2_2⁴ + c_2_1⁴·c_2_2² + c_2_1⁶, an element of degree 12
7. a_15_1 → c_2_1·c_2_2⁶·a_1_0 + c_2_1²·c_2_2⁵·a_1_1 − c_2_1³·c_2_2⁴·a_1_0
      − c_2_1⁶·c_2_2·a_1_1, an element of degree 15
8. b_16_1 → c_2_2⁸ + c_2_1²·c_2_2⁶ + c_2_1⁴·c_2_2⁴ + c_2_1⁶·c_2_2², an element of degree 16

About the group

Ring generators

Ring relations

Completion information

Restriction maps