Simon King
David J. Green
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Cohomology of group number 1730 of order 128
General information on the group
- The group has 4 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 3.
- It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 4.
- The depth exceeds the Duflot bound, which is 3.
- The Poincaré series is
- The a-invariants are -∞,-∞,-∞,-∞,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 8 minimal generators of maximal degree 4:
- a_1_1, a nilpotent element of degree 1
- b_1_0, an element of degree 1
- b_1_2, an element of degree 1
- c_1_3, a Duflot regular element of degree 1
- a_2_8, a nilpotent element of degree 2
- c_2_9, a Duflot regular element of degree 2
- b_3_19, an element of degree 3
- c_4_34, a Duflot regular element of degree 4
Ring relations
There are 9 minimal relations of maximal degree 6:
- a_1_1·b_1_0
- b_1_0·b_1_2
- a_1_13
- a_2_8·b_1_0
- a_1_12·b_1_22 + a_2_8·a_1_1·b_1_2 + a_2_82 + c_2_9·a_1_12
- b_1_2·b_3_19 + a_2_8·a_1_12
- a_1_1·b_3_19
- a_2_8·b_3_19
- b_3_192 + c_4_34·b_1_02
Data used for Benson′s test
- Benson′s completion test succeeded in degree 6.
- 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_1_3, a Duflot regular element of degree 1
- c_2_9, a Duflot regular element of degree 2
- c_4_34, a Duflot regular element of degree 4
- b_1_22 + b_1_02, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 5].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 3
- a_1_1 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- a_2_8 → 0, an element of degree 2
- c_2_9 → c_1_12, an element of degree 2
- b_3_19 → 0, an element of degree 3
- c_4_34 → c_1_24, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_1 → 0, an element of degree 1
- b_1_0 → c_1_3, an element of degree 1
- b_1_2 → 0, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- a_2_8 → 0, an element of degree 2
- c_2_9 → c_1_1·c_1_3 + c_1_12, an element of degree 2
- b_3_19 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
- c_4_34 → c_1_22·c_1_32 + c_1_24, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_1 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- a_2_8 → 0, an element of degree 2
- c_2_9 → c_1_1·c_1_3 + c_1_12, an element of degree 2
- b_3_19 → 0, an element of degree 3
- c_4_34 → c_1_24, an element of degree 4
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