Simon King
David J. Green
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Singular
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Cohomology of group number 2161 of order 128
General information on the group
- The group has 5 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 2.
- It has 6 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 4.
- The depth exceeds the Duflot bound, which is 2.
- The Poincaré series is
t2 + t + 1 |
| (t − 1)4 · (t2 + 1) |
- The a-invariants are -∞,-∞,-∞,-∞,-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_1_4, an element of degree 1
- c_2_13, a Duflot regular element of degree 2
- c_4_50, a Duflot regular element of degree 4
Ring relations
There are 3 minimal relations of maximal degree 3:
- a_1_02
- b_1_3·b_1_4 + b_1_1·b_1_4 + b_1_1·b_1_2 + a_1_0·b_1_3 + a_1_0·b_1_1
- b_1_1·b_1_2·b_1_4 + b_1_1·b_1_2·b_1_3 + b_1_1·b_1_22 + a_1_0·b_1_1·b_1_2
Data used for Benson′s test
- Benson′s completion test succeeded in degree 6.
- However, the last relation was already found in degree 3 and the last generator in degree 4.
- The following is a filter regular homogeneous system of parameters:
- c_2_13, a Duflot regular element of degree 2
- c_4_50, a Duflot regular element of degree 4
- b_1_42 + b_1_32 + b_1_2·b_1_4 + b_1_2·b_1_3 + b_1_22 + b_1_1·b_1_4 + b_1_1·b_1_3
+ b_1_12, an element of degree 2
- b_1_42 + b_1_32, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 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_1_4 → 0, an element of degree 1
- c_2_13 → c_1_12, an element of degree 2
- c_4_50 → c_1_04, 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_1_4 → 0, an element of degree 1
- c_2_13 → c_1_12, an element of degree 2
- c_4_50 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3
+ c_1_02·c_1_22 + c_1_04, 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_1_4 → c_1_3, an element of degree 1
- c_2_13 → c_1_32 + c_1_12, an element of degree 2
- c_4_50 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3
+ c_1_02·c_1_22 + c_1_04, 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 → 0, 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_1_4 → 0, an element of degree 1
- c_2_13 → c_1_12, an element of degree 2
- c_4_50 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3
+ c_1_02·c_1_22 + c_1_04, 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 → 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_1_4 → c_1_3, an element of degree 1
- c_2_13 → c_1_32 + c_1_12, an element of degree 2
- c_4_50 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3
+ c_1_02·c_1_22 + c_1_04, 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, 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_1_4 → c_1_2, an element of degree 1
- c_2_13 → c_1_22 + c_1_12, an element of degree 2
- c_4_50 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3
+ c_1_02·c_1_22 + c_1_04, 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, an element of degree 1
- b_1_2 → c_1_3 + c_1_2, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- b_1_4 → c_1_3, an element of degree 1
- c_2_13 → c_1_32 + c_1_12, an element of degree 2
- c_4_50 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3
+ c_1_02·c_1_22 + c_1_04, an element of degree 4
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