Simon King
David J. Green
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Singular
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Cohomology of group number 151 of order 128
General information on the group
- The group has 2 minimal generators and exponent 32.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 1.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 3 and depth 1.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t2 − t + 1) |
| (t − 1)3 · (t2 + 1) |
- The a-invariants are -∞,-3,-3,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 8 minimal generators of maximal degree 4:
- a_1_0, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- b_2_1, an element of degree 2
- b_2_2, an element of degree 2
- b_3_3, an element of degree 3
- b_3_4, an element of degree 3
- b_4_6, an element of degree 4
- c_4_7, a Duflot regular element of degree 4
Ring relations
There are 18 minimal relations of maximal degree 8:
- a_1_02
- a_1_0·b_1_1
- b_2_1·a_1_0
- b_2_2·a_1_0
- b_2_2·b_1_12 + b_2_12
- a_1_0·b_3_3
- b_1_1·b_3_3 + b_2_1·b_2_2
- a_1_0·b_3_4
- b_2_22·b_1_1 + b_2_1·b_3_3
- b_4_6·a_1_0
- b_4_6·b_1_1 + b_2_1·b_3_4 + b_2_1·b_2_2·b_1_1
- b_3_32 + b_2_23
- b_3_42 + b_2_2·b_1_1·b_3_4 + c_4_7·b_1_12
- b_2_2·b_1_1·b_3_4 + b_2_1·b_4_6 + b_2_12·b_2_2
- b_3_3·b_3_4 + b_2_2·b_4_6 + b_2_1·b_2_22
- b_4_6·b_3_3 + b_2_22·b_3_4 + b_2_1·b_2_2·b_3_3
- b_4_6·b_3_4 + b_2_1·c_4_7·b_1_1
- b_4_62 + b_2_1·b_2_2·b_4_6 + b_2_12·c_4_7
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_7, a Duflot regular element of degree 4
- b_1_12 + b_2_2 + b_2_1, an element of degree 2
- b_1_12, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, 1, 3, 5].
- 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 1
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_2_1 → 0, an element of degree 2
- b_2_2 → 0, an element of degree 2
- b_3_3 → 0, an element of degree 3
- b_3_4 → 0, an element of degree 3
- b_4_6 → 0, an element of degree 4
- c_4_7 → c_1_04, an element of degree 4
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_2_1 → c_1_1·c_1_2, an element of degree 2
- b_2_2 → c_1_22, an element of degree 2
- b_3_3 → c_1_23, an element of degree 3
- b_3_4 → c_1_1·c_1_22 + c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
- b_4_6 → c_1_0·c_1_12·c_1_2 + c_1_02·c_1_1·c_1_2, an element of degree 4
- c_4_7 → c_1_0·c_1_1·c_1_22 + c_1_02·c_1_22 + c_1_02·c_1_12 + c_1_04, an element of degree 4
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