Simon King
David J. Green
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Singular
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Cohomology of group number 2148 of order 128
General information on the group
- The group has 4 minimal generators and exponent 16.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 1.
- It has 4 conjugacy classes of maximal elementary abelian subgroups, which are of rank 2, 3, 3 and 3, respectively.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 3 and depth 2.
- The depth exceeds the Duflot bound, which is 1.
- The Poincaré series is
( − 1) · (t2 + t + 1) |
| (t − 1)3 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-∞,-3,-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 8:
- 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_5_26, an element of degree 5
- c_8_45, a Duflot regular element of degree 8
Ring relations
There are 5 minimal relations of maximal degree 10:
- b_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 + b_1_0·b_1_2·b_1_3 + b_1_03
- b_1_03·b_1_32 + b_1_03·b_1_2·b_1_3 + b_1_03·b_1_22 + b_1_04·b_1_3
+ b_1_04·b_1_2 + b_1_05
- b_1_0·b_5_26 + b_1_02·b_1_23·b_1_3 + b_1_03·b_1_22·b_1_3 + b_1_04·b_1_2·b_1_3
+ b_1_04·b_1_22 + b_1_05·b_1_3 + b_1_06
- b_5_262 + b_1_12·b_1_23·b_5_26 + b_1_13·b_1_2·b_1_3·b_5_26
+ b_1_13·b_1_26·b_1_3 + b_1_15·b_1_24·b_1_3 + b_1_16·b_1_23·b_1_3 + b_1_17·b_1_22·b_1_3 + b_1_02·b_1_27·b_1_3 + b_1_07·b_1_22·b_1_3 + b_1_08·b_1_2·b_1_3 + b_1_09·b_1_3 + c_8_45·b_1_12
Data used for Benson′s test
- Benson′s completion test succeeded in degree 10.
- 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_8_45, a Duflot regular element of degree 8
- 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_1·b_1_32 + b_1_1·b_1_2·b_1_3 + b_1_1·b_1_22 + b_1_12·b_1_3 + b_1_12·b_1_2
+ b_1_0·b_1_32 + b_1_02·b_1_2, an element of degree 3
- The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 10].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
- We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 2 elements of degree 2.
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 1
- 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_5_26 → 0, an element of degree 5
- c_8_45 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 2
- b_1_0 → c_1_1, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_1, an element of degree 1
- b_1_3 → c_1_1, an element of degree 1
- b_5_26 → 0, an element of degree 5
- c_8_45 → c_1_04·c_1_14 + c_1_08, an element of degree 8
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_1, 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_5_26 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
+ c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- c_8_45 → c_1_0·c_1_12·c_1_25 + c_1_0·c_1_13·c_1_24 + c_1_02·c_1_1·c_1_25
+ c_1_02·c_1_13·c_1_23 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24 + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8
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_1, 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_5_26 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
+ c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- c_8_45 → c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24
+ c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8
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 + c_1_1, an element of degree 1
- b_1_2 → c_1_1, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- b_5_26 → c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_0·c_1_1·c_1_23 + c_1_0·c_1_13·c_1_2
+ c_1_02·c_1_23 + c_1_02·c_1_13 + c_1_04·c_1_2 + c_1_04·c_1_1, an element of degree 5
- c_8_45 → c_1_12·c_1_26 + c_1_14·c_1_24 + c_1_16·c_1_22 + c_1_0·c_1_12·c_1_25
+ c_1_0·c_1_13·c_1_24 + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16 + c_1_04·c_1_24 + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_12·c_1_22 + c_1_08, an element of degree 8
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