Simon King
David J. Green
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Cohomology of group number 1629 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 2.
- It has 3 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 3.
- The depth exceeds the Duflot bound, which is 2.
- The Poincaré series is
t2 + t + 1 |
| (t + 1) · (t − 1)4 · (t2 + 1) |
- The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 7 minimal generators of maximal degree 4:
- a_1_0, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- b_1_3, an element of degree 1
- b_2_8, an element of degree 2
- c_2_9, a Duflot regular element of degree 2
- c_4_31, a Duflot regular element of degree 4
Ring relations
There are 5 minimal relations of maximal degree 4:
- a_1_02
- a_1_0·b_1_1
- b_1_2·b_1_32 + b_1_22·b_1_3 + b_1_1·b_1_2·b_1_3 + a_1_0·b_1_2·b_1_3
- b_2_8·a_1_0
- b_2_82 + c_2_9·b_1_12
Data used for Benson′s test
- Benson′s completion test succeeded in degree 6.
- However, the last relation was already found in degree 4 and the last generator in degree 4.
- The following is a filter regular homogeneous system of parameters:
- c_2_9, a Duflot regular element of degree 2
- c_4_31, a Duflot regular element of degree 4
- b_1_32 + b_1_2·b_1_3 + b_1_22 + b_1_1·b_1_3 + b_1_1·b_1_2 + b_1_12, an element of degree 2
- b_1_12, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 4, 6].
- 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 2
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_2_8 → 0, an element of degree 2
- c_2_9 → c_1_02, an element of degree 2
- c_4_31 → c_1_14, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_2_8 → c_1_0·c_1_2, an element of degree 2
- c_2_9 → c_1_02, an element of degree 2
- c_4_31 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3
+ c_1_12·c_1_22 + c_1_14, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → c_1_3, an element of degree 1
- b_2_8 → c_1_0·c_1_2, an element of degree 2
- c_2_9 → c_1_02, an element of degree 2
- c_4_31 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3
+ c_1_12·c_1_22 + c_1_14, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_3 + c_1_2, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_1_3 → c_1_3, an element of degree 1
- b_2_8 → c_1_0·c_1_3 + c_1_0·c_1_2, an element of degree 2
- c_2_9 → c_1_02, an element of degree 2
- c_4_31 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3
+ c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_14, an element of degree 4
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