Simon King
David J. Green
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Cohomology of group number 1989 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 2.
- It has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 3 and depth 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t5 + 2·t4 + 2·t2 + 2·t + 1) |
| (t + 1) · (t − 1)3 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-5,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 10 minimal generators of maximal degree 4:
- b_1_0, an 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_4_8, an element of degree 4
- b_4_9, an element of degree 4
- b_4_10, an element of degree 4
- b_4_11, an element of degree 4
- c_4_12, a Duflot regular element of degree 4
- c_4_13, a Duflot regular element of degree 4
Ring relations
There are 23 minimal relations of maximal degree 8:
- b_1_0·b_1_2
- b_1_32 + b_1_1·b_1_2 + b_1_0·b_1_3 + b_1_0·b_1_1
- b_1_1·b_1_22 + b_1_12·b_1_2 + b_1_0·b_1_12
- b_1_1·b_1_2·b_1_3 + b_1_1·b_1_22 + b_1_12·b_1_3
- b_4_8·b_1_2
- b_4_9·b_1_2
- b_1_03·b_1_1·b_1_3 + b_4_9·b_1_3 + b_4_8·b_1_3 + b_4_8·b_1_1
- b_1_03·b_1_1·b_1_3 + b_4_9·b_1_0 + b_4_8·b_1_3
- b_4_10·b_1_0 + b_4_8·b_1_3 + b_4_8·b_1_1 + b_4_8·b_1_0
- b_1_0·b_1_14 + b_4_10·b_1_1 + b_4_9·b_1_1 + b_4_8·b_1_1
- b_1_0·b_1_14 + b_4_11·b_1_2 + b_4_10·b_1_3 + b_4_9·b_1_1 + b_4_8·b_1_1
- b_4_11·b_1_3 + b_4_8·b_1_3
- b_1_0·b_1_14 + b_4_11·b_1_0 + b_4_9·b_1_1 + b_4_8·b_1_1 + b_4_8·b_1_0
- b_4_9·b_1_04 + b_4_8·b_1_02·b_1_1·b_1_3 + b_4_82 + c_4_13·b_1_04
+ c_4_12·b_1_03·b_1_3 + c_4_12·b_1_03·b_1_1
- b_4_92 + b_4_8·b_1_02·b_1_1·b_1_3 + b_4_8·b_1_03·b_1_1 + b_4_82
+ c_4_13·b_1_03·b_1_3 + c_4_13·b_1_03·b_1_1 + c_4_13·b_1_04
- b_4_8·b_1_02·b_1_1·b_1_3 + b_4_8·b_1_03·b_1_1 + b_4_8·b_4_9 + b_4_82
+ c_4_13·b_1_03·b_1_3 + c_4_13·b_1_04 + c_4_12·b_1_02·b_1_1·b_1_3
- b_4_102 + b_4_8·b_1_02·b_1_1·b_1_3 + b_4_8·b_1_03·b_1_1 + c_4_13·b_1_24
+ c_4_13·b_1_03·b_1_3 + c_4_13·b_1_03·b_1_1 + c_4_13·b_1_04 + c_4_12·b_1_24
- b_4_9·b_4_10 + c_4_13·b_1_02·b_1_1·b_1_3 + c_4_13·b_1_03·b_1_1
- b_4_8·b_1_02·b_1_1·b_1_3 + b_4_8·b_1_03·b_1_1 + b_4_8·b_4_10 + c_4_13·b_1_03·b_1_3
+ c_4_13·b_1_03·b_1_1 + c_4_13·b_1_04
- b_4_11·b_1_14 + b_4_112 + b_4_9·b_1_14 + b_4_82 + c_4_12·b_1_14
- b_4_10·b_1_14 + b_4_10·b_4_11 + b_4_8·b_1_02·b_1_1·b_1_3 + b_4_8·b_1_03·b_1_1
+ c_4_13·b_1_23·b_1_3 + c_4_13·b_1_02·b_1_1·b_1_3 + c_4_13·b_1_03·b_1_3 + c_4_13·b_1_04 + c_4_12·b_1_23·b_1_3 + c_4_12·b_1_0·b_1_13
- b_4_9·b_4_11 + b_4_8·b_1_02·b_1_1·b_1_3 + b_4_8·b_1_03·b_1_1 + b_4_82
+ c_4_13·b_1_03·b_1_3 + c_4_13·b_1_04 + c_4_12·b_1_0·b_1_13 + c_4_12·b_1_02·b_1_1·b_1_3
- b_4_8·b_4_11 + b_4_82 + c_4_13·b_1_02·b_1_1·b_1_3 + c_4_13·b_1_03·b_1_1
Data used for Benson′s test
- Benson′s completion test succeeded in degree 8.
- 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_4_12, a Duflot regular element of degree 4
- c_4_13, a Duflot regular element of degree 4
- 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, 3, 7].
- 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 2
- 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
- b_1_3 → 0, an element of degree 1
- b_4_8 → 0, an element of degree 4
- b_4_9 → 0, an element of degree 4
- b_4_10 → 0, an element of degree 4
- b_4_11 → 0, an element of degree 4
- c_4_12 → c_1_04, an element of degree 4
- c_4_13 → c_1_14 + c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → c_1_2, 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_4_8 → c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
- b_4_9 → 0, an element of degree 4
- b_4_10 → c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
- b_4_11 → c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
- c_4_12 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_13 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 3
- b_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 → 0, an element of degree 1
- b_4_8 → 0, an element of degree 4
- b_4_9 → 0, an element of degree 4
- b_4_10 → 0, an element of degree 4
- b_4_11 → c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
- c_4_12 → c_1_0·c_1_23 + c_1_04, an element of degree 4
- c_4_13 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_4_8 → 0, an element of degree 4
- b_4_9 → 0, an element of degree 4
- b_4_10 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
- b_4_11 → 0, an element of degree 4
- c_4_12 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_13 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → c_1_2, 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 → c_1_2, an element of degree 1
- b_4_8 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
- b_4_9 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
- b_4_10 → 0, an element of degree 4
- b_4_11 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
- c_4_12 → c_1_24 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_13 → c_1_24 + c_1_1·c_1_23 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
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