Simon King
David J. Green
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Cohomology of group number 90 of order 128
General information on the group
- The group has 2 minimal generators and exponent 16.
- 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
t5 + t2 + 1 |
| (t − 1)2 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-2,-2. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 9 minimal generators of maximal degree 8:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- a_2_1, a nilpotent element of degree 2
- b_2_2, an element of degree 2
- a_3_3, a nilpotent element of degree 3
- a_5_2, a nilpotent element of degree 5
- a_5_4, a nilpotent element of degree 5
- a_6_4, a nilpotent element of degree 6
- c_8_6, a Duflot regular element of degree 8
Ring relations
There are 27 minimal relations of maximal degree 12:
- a_1_02
- a_1_0·a_1_1
- a_2_1·a_1_0
- b_2_2·a_1_0 + a_1_13
- a_2_12 + a_2_1·a_1_12
- a_2_1·b_2_2 + a_1_1·a_3_3
- a_1_0·a_3_3 + a_2_12
- b_2_2·a_1_13
- a_2_1·a_3_3 + a_1_12·a_3_3
- a_3_32 + b_2_2·a_1_1·a_3_3 + a_1_13·a_3_3
- a_1_1·a_5_2
- a_3_32 + a_1_0·a_5_2 + b_2_2·a_1_1·a_3_3
- a_1_0·a_5_4
- a_2_1·a_5_2
- b_2_2·a_5_2 + a_1_12·a_5_4
- a_6_4·a_1_1 + a_2_1·a_5_4 + b_2_2·a_1_12·a_3_3
- a_6_4·a_1_0
- a_3_3·a_5_2 + a_2_1·a_1_1·a_5_4
- b_2_2·a_6_4 + a_3_3·a_5_4 + b_2_22·a_1_1·a_3_3
- a_3_3·a_5_2 + a_2_1·a_6_4
- a_6_4·a_3_3 + a_1_1·a_3_3·a_5_4 + b_2_22·a_1_12·a_3_3
- a_5_22
- a_5_2·a_5_4
- a_5_42 + b_2_22·a_1_1·a_5_4 + c_8_6·a_1_12
- a_6_4·a_5_2
- a_6_4·a_5_4 + a_2_1·c_8_6·a_1_1
- a_6_42 + b_2_2·a_1_12·a_3_3·a_5_4 + a_2_1·c_8_6·a_1_12
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:
- c_8_6, a Duflot regular element of degree 8
- b_2_2, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, 6, 8].
- The filter degree type of any filter regular HSOP is [-1, -2, -2].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 1
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- a_2_1 → 0, an element of degree 2
- b_2_2 → 0, an element of degree 2
- a_3_3 → 0, an element of degree 3
- a_5_2 → 0, an element of degree 5
- a_5_4 → 0, an element of degree 5
- a_6_4 → 0, an element of degree 6
- c_8_6 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 2
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- a_2_1 → 0, an element of degree 2
- b_2_2 → c_1_12, an element of degree 2
- a_3_3 → 0, an element of degree 3
- a_5_2 → 0, an element of degree 5
- a_5_4 → 0, an element of degree 5
- a_6_4 → 0, an element of degree 6
- c_8_6 → c_1_04·c_1_14 + c_1_08, an element of degree 8
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