Simon King
David J. Green
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Cohomology of group number 1909 of order 128
General information on the group
- The group has 4 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 2.
- 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 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
t4 + t3 − t2 − t − 1 |
| (t − 1)3 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 7 minimal generators of maximal degree 5:
- a_1_0, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- a_1_3, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- c_4_8, a Duflot regular element of degree 4
- c_4_9, a Duflot regular element of degree 4
- b_5_12, an element of degree 5
Ring relations
There are 7 minimal relations of maximal degree 10:
- a_1_22 + a_1_0·a_1_2
- a_1_0·b_1_1 + a_1_32
- a_1_3·b_1_12 + a_1_33 + a_1_03
- a_1_32·b_1_1 + a_1_0·a_1_32
- a_1_0·b_5_12 + c_4_9·a_1_32 + c_4_9·a_1_0·a_1_2 + c_4_9·a_1_02 + c_4_8·a_1_32
+ c_4_8·a_1_0·a_1_2
- a_1_3·b_5_12 + c_4_9·a_1_3·b_1_1 + c_4_8·a_1_3·b_1_1 + c_4_9·a_1_2·a_1_3
+ c_4_9·a_1_0·a_1_3 + c_4_8·a_1_2·a_1_3
- b_5_122 + c_4_8·b_1_16 + c_4_92·b_1_12 + c_4_82·b_1_12 + c_4_92·a_1_0·a_1_2
+ c_4_92·a_1_02 + c_4_82·a_1_0·a_1_2
Data used for Benson′s test
- Benson′s completion test succeeded in degree 10.
- 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_8, a Duflot regular element of degree 4
- c_4_9, a Duflot regular element of degree 4
- b_1_12, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 7].
- 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 2
- a_1_0 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- c_4_8 → c_1_04, an element of degree 4
- c_4_9 → c_1_14 + c_1_04, an element of degree 4
- b_5_12 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- c_4_8 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_9 → c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_23 + c_1_04, an element of degree 4
- b_5_12 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
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