Simon King
David J. Green
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Cohomology of group number 176 of order 128
General information on the group
- The group has 3 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 4.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 4.
- The depth coincides with the Duflot bound.
- The Poincaré series is
t2 + t + 1 |
| (t + 1)2 · (t − 1)4 |
- The a-invariants are -∞,-∞,-∞,-∞,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 9 minimal generators of maximal degree 2:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- a_2_2, a nilpotent element of degree 2
- a_2_3, a nilpotent element of degree 2
- c_2_4, a Duflot regular element of degree 2
- c_2_5, a Duflot regular element of degree 2
- c_2_6, a Duflot regular element of degree 2
- c_2_7, a Duflot regular element of degree 2
Ring relations
There are 10 minimal relations of maximal degree 4:
- a_1_02
- a_1_12 + a_1_0·a_1_1
- a_1_1·a_1_2
- a_1_22 + a_1_0·a_1_2
- a_2_2·a_1_1
- a_2_3·a_1_1 + a_2_3·a_1_0 + a_2_2·a_1_2 + a_2_2·a_1_0
- a_2_3·a_1_2
- a_2_22 + a_2_2·a_1_0·a_1_2 + c_2_6·a_1_0·a_1_1 + c_2_5·a_1_0·a_1_2
- a_2_2·a_2_3 + c_2_6·a_1_0·a_1_2 + c_2_6·a_1_0·a_1_1
- a_2_32 + a_2_2·a_1_0·a_1_2 + c_2_7·a_1_0·a_1_1 + c_2_6·a_1_0·a_1_2
Data used for Benson′s test
- Benson′s completion test succeeded in degree 4.
- 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_2_4, a Duflot regular element of degree 2
- c_2_5, a Duflot regular element of degree 2
- c_2_6, a Duflot regular element of degree 2
- c_2_7, a Duflot regular element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 4].
- 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 4
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- a_2_2 → 0, an element of degree 2
- a_2_3 → 0, an element of degree 2
- c_2_4 → c_1_32, an element of degree 2
- c_2_5 → c_1_02, an element of degree 2
- c_2_6 → c_1_22, an element of degree 2
- c_2_7 → c_1_12, an element of degree 2
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