Simon King
David J. Green
Cohomology
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Cohomology of group number 1690 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 3.
- Its center has rank 3.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 3 and depth 3.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t2 + t + 1) |
| (t − 1)3 · (t2 + 1) |
- The a-invariants are -∞,-∞,-∞,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 6 minimal generators of maximal degree 4:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- c_1_3, a Duflot regular element of degree 1
- c_2_8, a Duflot regular element of degree 2
- c_4_20, a Duflot regular element of degree 4
Ring relations
There are 3 minimal relations of maximal degree 3:
- a_1_12 + a_1_0·a_1_1 + a_1_02
- a_1_22 + a_1_02
- a_1_03
Data used for Benson′s test
- Benson′s completion test succeeded in degree 5.
- However, the last relation was already found in degree 3 and the last generator in degree 4.
- The following is a filter regular homogeneous system of parameters:
- c_1_3, a Duflot regular element of degree 1
- c_2_8, a Duflot regular element of degree 2
- c_4_20, a Duflot regular element of degree 4
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 4].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 3
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- c_2_8 → c_1_22, an element of degree 2
- c_4_20 → c_1_14, an element of degree 4
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