Simon King
David J. Green
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Singular
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Cohomology of group number 2305 of order 128
General information on the group
- The group has 5 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 1.
- It has 15 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 3.
- The depth exceeds the Duflot bound, which is 1.
- The Poincaré series is
( − 1) · (t2 + t + 1) · (t4 + t3 + t2 + t + 1) |
| (t − 1)3 · (t2 + 1) · (t4 + 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 8:
- 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_1_4, an element of degree 1
- c_8_165, a Duflot regular element of degree 8
Ring relations
There are 3 minimal relations of maximal degree 5:
- a_1_02
- b_1_2·b_1_32 + b_1_22·b_1_3 + b_1_1·b_1_42 + b_1_12·b_1_4
- b_1_1·b_1_44 + b_1_1·b_1_32·b_1_42 + b_1_1·b_1_2·b_1_3·b_1_42
+ b_1_1·b_1_22·b_1_42 + b_1_12·b_1_32·b_1_4 + b_1_12·b_1_2·b_1_3·b_1_4 + b_1_12·b_1_22·b_1_4 + b_1_14·b_1_4
Data used for Benson′s test
- Benson′s completion test succeeded in degree 10.
- However, the last relation was already found in degree 5 and the last generator in degree 8.
- The following is a filter regular homogeneous system of parameters:
- c_8_165, a Duflot regular element of degree 8
- b_1_44 + b_1_32·b_1_42 + b_1_34 + b_1_2·b_1_3·b_1_42 + b_1_22·b_1_42
+ b_1_23·b_1_3 + b_1_24 + b_1_1·b_1_32·b_1_4 + b_1_1·b_1_2·b_1_42 + b_1_1·b_1_2·b_1_3·b_1_4 + b_1_1·b_1_22·b_1_4 + b_1_12·b_1_42 + b_1_12·b_1_32 + b_1_12·b_1_2·b_1_4 + b_1_12·b_1_2·b_1_3 + b_1_12·b_1_22 + b_1_14, an element of degree 4
- b_1_32·b_1_44 + b_1_34·b_1_42 + b_1_2·b_1_3·b_1_44 + b_1_22·b_1_44
+ b_1_23·b_1_3·b_1_42 + b_1_24·b_1_42 + b_1_1·b_1_34·b_1_4 + b_1_1·b_1_23·b_1_42 + b_1_1·b_1_23·b_1_3·b_1_4 + b_1_1·b_1_24·b_1_4 + b_1_12·b_1_32·b_1_42 + b_1_12·b_1_34 + b_1_12·b_1_2·b_1_3·b_1_42 + b_1_12·b_1_22·b_1_42 + b_1_12·b_1_23·b_1_4 + b_1_12·b_1_23·b_1_3 + b_1_12·b_1_24 + b_1_14·b_1_32 + b_1_14·b_1_2·b_1_3 + b_1_14·b_1_22, an element of degree 6
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 15].
- 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
- 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_1_4 → 0, an element of degree 1
- c_8_165 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_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_1_4 → 0, an element of degree 1
- c_8_165 → 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
- a_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_1_4 → 0, an element of degree 1
- c_8_165 → 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
- a_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 → c_1_2, an element of degree 1
- b_1_4 → 0, an element of degree 1
- c_8_165 → c_1_12·c_1_26 + c_1_13·c_1_25 + 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
- a_1_0 → 0, 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 → 0, an element of degree 1
- b_1_4 → c_1_2, an element of degree 1
- c_8_165 → 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
- 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_1, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_1_4 → c_1_2, an element of degree 1
- c_8_165 → c_1_28 + c_1_1·c_1_27 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24
+ 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
- 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 → c_1_1, an element of degree 1
- b_1_4 → c_1_2, an element of degree 1
- c_8_165 → 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
- 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_1, an element of degree 1
- b_1_4 → c_1_2, an element of degree 1
- c_8_165 → c_1_28 + c_1_13·c_1_25 + 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
- a_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 → c_1_2, an element of degree 1
- b_1_4 → c_1_1, an element of degree 1
- c_8_165 → c_1_1·c_1_27 + c_1_13·c_1_25 + 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
- a_1_0 → 0, an element of degree 1
- b_1_1 → 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_1_4 → c_1_2, an element of degree 1
- c_8_165 → c_1_13·c_1_25 + c_1_16·c_1_22 + c_1_17·c_1_2 + 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
- a_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_1_4 → c_1_2, an element of degree 1
- c_8_165 → c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·c_1_23 + c_1_17·c_1_2
+ 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
- a_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 → c_1_1, an element of degree 1
- b_1_4 → c_1_2, an element of degree 1
- c_8_165 → c_1_1·c_1_27 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_17·c_1_2
+ 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
- a_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_2, an element of degree 1
- b_1_3 → c_1_1, an element of degree 1
- b_1_4 → c_1_2, an element of degree 1
- c_8_165 → c_1_28 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22
+ c_1_17·c_1_2 + 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
- 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_2 + c_1_1, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- b_1_4 → c_1_1, an element of degree 1
- c_8_165 → c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22 + c_1_17·c_1_2
+ 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
- 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_2, an element of degree 1
- b_1_3 → c_1_2 + c_1_1, an element of degree 1
- b_1_4 → c_1_1, an element of degree 1
- c_8_165 → c_1_1·c_1_27 + c_1_13·c_1_25 + c_1_16·c_1_22 + 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
- a_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 → c_1_2, an element of degree 1
- b_1_4 → c_1_1, an element of degree 1
- c_8_165 → c_1_1·c_1_27 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_17·c_1_2 + c_1_18
+ 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
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