Simon King
David J. Green
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Cohomology of group number 1729 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 5.
- Its center has rank 3.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 5.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 5 and depth 3.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t2 − t + 1) |
| (t − 1)5 · (t2 + 1) |
- The a-invariants are -∞,-∞,-∞,-5,-5,-5. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 10 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
- c_1_3, a Duflot regular element of degree 1
- b_2_8, an element of degree 2
- c_2_9, a Duflot regular element of degree 2
- b_3_19, an element of degree 3
- b_3_20, an element of degree 3
- b_4_37, an element of degree 4
- c_4_39, a Duflot regular element of degree 4
Ring relations
There are 18 minimal relations of maximal degree 8:
- a_1_0·b_1_1
- a_1_0·b_1_2
- a_1_03
- b_2_8·a_1_0
- b_2_8·b_1_1·b_1_2 + b_2_82 + c_2_9·b_1_12
- a_1_0·b_3_19
- a_1_0·b_3_20
- b_1_2·b_3_19 + b_1_1·b_3_20
- b_4_37·b_1_2 + b_2_8·b_3_20
- b_4_37·a_1_0
- b_4_37·b_1_1 + b_2_8·b_3_19
- b_3_202 + c_4_39·b_1_22
- b_3_19·b_3_20 + c_4_39·b_1_1·b_1_2
- b_3_192 + c_4_39·b_1_12
- b_2_8·b_1_1·b_3_20 + b_2_8·b_4_37 + c_2_9·b_1_1·b_3_19
- b_4_37·b_3_19 + b_2_8·c_4_39·b_1_1
- b_4_37·b_3_20 + b_2_8·c_4_39·b_1_2
- b_4_372 + b_2_82·c_4_39
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_1_3, a Duflot regular element of degree 1
- c_2_9, a Duflot regular element of degree 2
- c_4_39, a Duflot regular element of degree 4
- b_1_22 + b_1_1·b_1_2 + b_1_12, an element of degree 2
- b_1_22, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 2, 4, 6].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 3
- 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
- c_1_3 → c_1_0, an element of degree 1
- b_2_8 → 0, an element of degree 2
- c_2_9 → c_1_12, an element of degree 2
- b_3_19 → 0, an element of degree 3
- b_3_20 → 0, an element of degree 3
- b_4_37 → 0, an element of degree 4
- c_4_39 → c_1_24, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 5
- 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_4, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- b_2_8 → c_1_1·c_1_3, an element of degree 2
- c_2_9 → c_1_1·c_1_4 + c_1_12, an element of degree 2
- b_3_19 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
- b_3_20 → c_1_2·c_1_3·c_1_4 + c_1_22·c_1_4, an element of degree 3
- b_4_37 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3, an element of degree 4
- c_4_39 → c_1_22·c_1_32 + c_1_24, an element of degree 4
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