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

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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 2-Rank 3.
• The centre of a Sylow 2-subgroup has rank 1.
• Its Sylow 2-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 2 and 3, respectively.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 3.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  ( − 1)·(1  −  t  +  t2  −  t3  +  t4)

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

• The a-invariants are -∞,-∞,-∞,-3. 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, -3, -3].

About the group

Ring generators

Ring relations

Completion information

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Ring generators

The cohomology ring has 4 minimal generators of maximal degree 5:

1. b_1_0, an element of degree 1
2. b_3_0, an element of degree 3
3. c_4_2, a Duflot element of degree 4
4. b_5_3, an element of degree 5

About the group

Ring generators

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Ring relations

There is one minimal relation of degree 10:

1. b_5_3² + b_1_0·b_3_0³ + b_1_0²·b_3_0·b_5_3 + c_4_2·b_3_0²

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 10 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. b_1_0⁴ + c_4_2, an element of degree 4
  2. b_3_0² + b_1_0·b_5_3 + c_4_2·b_1_0², an element of degree 6
  3. b_1_0, an element of degree 1
• A Duflot regular sequence is given by c_4_2.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 8].
• Modifying the above filter regular HSOP, we obtained the following parameters:
1. b_1_0⁴ + c_4_2, an element of degree 4
2. b_3_0, an element of degree 3
3. b_1_0, an element of degree 1

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(96,193); GF(2))

1. b_1_0 → b_1_1 + b_1_0
2. b_3_0 → b_3_0 + b_1_0·b_1_1² + b_1_0²·b_1_1
3. c_4_2 → b_1_0·b_3_0 + b_1_0²·b_1_1² + c_4_7
4. b_5_3 → b_1_0²·b_1_1³ + b_1_0³·b_1_1² + c_4_7·b_1_1

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

1. b_1_0 → 0, an element of degree 1
2. b_3_0 → 0, an element of degree 3
3. c_4_2 → c_1_0⁴, an element of degree 4
4. b_5_3 → 0, an element of degree 5

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

1. b_1_0 → c_1_1, an element of degree 1
2. b_3_0 → 0, an element of degree 3
3. c_4_2 → c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
4. b_5_3 → 0, an element of degree 5

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

1. b_1_0 → c_1_2 + c_1_1, an element of degree 1
2. b_3_0 → c_1_1·c_1_2² + c_1_1²·c_1_2 + c_1_0·c_1_1² + c_1_0²·c_1_1, an element of degree 3
3. c_4_2 → c_1_1²·c_1_2² + c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2 + c_1_0²·c_1_2²
      + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
4. b_5_3 → c_1_1²·c_1_2³ + c_1_1³·c_1_2² + c_1_0·c_1_1²·c_1_2² + c_1_0²·c_1_1·c_1_2²
      + c_1_0²·c_1_1³ + c_1_0⁴·c_1_1, an element of degree 5

About the group

Ring generators

Ring relations

Completion information

Restriction maps