Simon King
David J. Green
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Cohomology of group number 1116 of order 128
General information on the group
- The group has 4 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 5.
- Its center has rank 4.
- It has 3 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 5.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 5 and depth 5.
- The depth exceeds the Duflot bound, which is 4.
- The Poincaré series is
( − 1) · (t2 + t + 1) |
| (t + 1)2 · (t − 1)5 |
- The a-invariants are -∞,-∞,-∞,-∞,-∞,-5. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 9 minimal generators of maximal degree 2:
- b_1_0, an element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- c_1_3, a Duflot regular element of degree 1
- b_2_7, an element of degree 2
- b_2_8, an element of degree 2
- c_2_9, a Duflot regular element of degree 2
- c_2_10, a Duflot regular element of degree 2
- c_2_11, a Duflot regular element of degree 2
Ring relations
There are 9 minimal relations of maximal degree 4:
- b_1_0·b_1_1
- b_1_0·b_1_2
- b_1_1·b_1_2
- b_2_7·b_1_1
- b_2_8·b_1_2 + b_2_7·b_1_2
- b_2_8·b_1_0
- b_2_72 + c_2_11·b_1_02 + c_2_9·b_1_22
- b_2_82 + c_2_10·b_1_12 + c_2_9·b_1_22
- b_2_7·b_2_8 + c_2_9·b_1_22
Data used for Benson′s test
- Benson′s completion test succeeded in degree 5.
- However, the last relation was already found in degree 4 and the last generator in degree 2.
- 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_2_10, a Duflot regular element of degree 2
- c_2_11, a Duflot regular element of degree 2
- b_1_22 + b_1_12 + b_1_02, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, -1, 4].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 4
- b_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
- c_1_3 → c_1_0, an element of degree 1
- b_2_7 → 0, an element of degree 2
- b_2_8 → 0, an element of degree 2
- c_2_9 → c_1_12, an element of degree 2
- c_2_10 → c_1_22, an element of degree 2
- c_2_11 → c_1_32, an element of degree 2
Restriction map to a maximal el. ab. subgp. of rank 5
- b_1_0 → c_1_4, an element of degree 1
- b_1_1 → 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
- b_2_7 → c_1_3·c_1_4, an element of degree 2
- b_2_8 → 0, an element of degree 2
- c_2_9 → c_1_1·c_1_4 + c_1_12, an element of degree 2
- c_2_10 → c_1_2·c_1_4 + c_1_22, an element of degree 2
- c_2_11 → c_1_32, an element of degree 2
Restriction map to a maximal el. ab. subgp. of rank 5
- b_1_0 → 0, an element of degree 1
- b_1_1 → c_1_4, 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
- b_2_7 → 0, an element of degree 2
- b_2_8 → c_1_2·c_1_4, an element of degree 2
- c_2_9 → c_1_1·c_1_4 + c_1_12, an element of degree 2
- c_2_10 → c_1_22, an element of degree 2
- c_2_11 → c_1_3·c_1_4 + c_1_32, an element of degree 2
Restriction map to a maximal el. ab. subgp. of rank 5
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_4, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- b_2_7 → c_1_1·c_1_4, an element of degree 2
- b_2_8 → c_1_1·c_1_4, an element of degree 2
- c_2_9 → c_1_12, an element of degree 2
- c_2_10 → c_1_2·c_1_4 + c_1_22, an element of degree 2
- c_2_11 → c_1_3·c_1_4 + c_1_32, an element of degree 2
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