Simon King
David J. Green
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Cohomology of group number 2320 of order 128
General information on the group
- The group is also known as D8xV16, the Direct product D8 x V_16.
- The group has 6 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 6.
- Its center has rank 5.
- It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 6.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 6 and depth 6.
- The depth exceeds the Duflot bound, which is 5.
- The Poincaré series is
- The a-invariants are -∞,-∞,-∞,-∞,-∞,-∞,-6. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 7 minimal generators of maximal degree 2:
- b_1_0, an element of degree 1
- b_1_1, an element of degree 1
- c_1_2, a Duflot regular element of degree 1
- c_1_3, a Duflot regular element of degree 1
- c_1_4, a Duflot regular element of degree 1
- c_1_5, a Duflot regular element of degree 1
- c_2_20, a Duflot regular element of degree 2
Ring relations
There is one minimal relation of degree 2:
- b_1_0·b_1_1
Data used for Benson′s test
- Benson′s completion test succeeded in degree 6.
- However, the last relation was already found in degree 2 and the last generator in degree 2.
- The following is a filter regular homogeneous system of parameters:
- c_1_2, a Duflot regular element of degree 1
- c_1_3, a Duflot regular element of degree 1
- c_1_4, a Duflot regular element of degree 1
- c_1_5, a Duflot regular element of degree 1
- c_2_20, a Duflot regular element of degree 2
- b_1_12 + b_1_02, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, -1, -1, 2].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -6, -6].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 5
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- c_1_2 → c_1_0, an element of degree 1
- c_1_3 → c_1_1, an element of degree 1
- c_1_4 → c_1_2, an element of degree 1
- c_1_5 → c_1_3, an element of degree 1
- c_2_20 → c_1_42, an element of degree 2
Restriction map to a maximal el. ab. subgp. of rank 6
- b_1_0 → c_1_5, an element of degree 1
- b_1_1 → 0, an element of degree 1
- c_1_2 → c_1_0, an element of degree 1
- c_1_3 → c_1_1, an element of degree 1
- c_1_4 → c_1_2, an element of degree 1
- c_1_5 → c_1_3, an element of degree 1
- c_2_20 → c_1_4·c_1_5 + c_1_42, an element of degree 2
Restriction map to a maximal el. ab. subgp. of rank 6
- b_1_0 → 0, an element of degree 1
- b_1_1 → c_1_5, an element of degree 1
- c_1_2 → c_1_0, an element of degree 1
- c_1_3 → c_1_1, an element of degree 1
- c_1_4 → c_1_2, an element of degree 1
- c_1_5 → c_1_3, an element of degree 1
- c_2_20 → c_1_4·c_1_5 + c_1_42, an element of degree 2
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