Simon King
David J. Green
Cohomology
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Cohomology of group number 1910 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 2 conjugacy classes of maximal elementary abelian subgroups, which are all 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
( − 1) · (t5 − 2·t4 + 2·t2 + 2·t + 1) |
| (t + 1) · (t − 1)3 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-4,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 9 minimal generators of maximal degree 5:
- a_1_2, a nilpotent element of degree 1
- a_1_3, a nilpotent element of degree 1
- b_1_0, an 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
- a_5_8, a nilpotent element of degree 5
- b_5_12, an element of degree 5
- b_5_13, an element of degree 5
Ring relations
There are 18 minimal relations of maximal degree 10:
- a_1_2·b_1_0 + a_1_22
- b_1_0·b_1_1 + a_1_32
- a_1_3·b_1_12 + a_1_2·b_1_12 + a_1_32·b_1_0 + a_1_33 + a_1_23
- a_1_32·b_1_1 + a_1_32·b_1_0
- a_1_3·a_5_8
- a_1_2·a_5_8
- b_1_1·b_5_12 + c_4_9·b_1_12 + c_4_9·a_1_3·b_1_1 + c_4_9·a_1_2·b_1_1
+ c_4_8·a_1_2·b_1_1 + c_4_9·a_1_32
- b_1_0·a_5_8 + a_1_3·b_5_12 + c_4_9·a_1_3·b_1_1 + c_4_9·a_1_3·b_1_0 + c_4_9·a_1_32
+ c_4_9·a_1_2·a_1_3 + c_4_8·a_1_2·a_1_3
- a_1_2·b_5_12 + c_4_9·a_1_2·b_1_1 + c_4_9·a_1_2·a_1_3 + c_4_8·a_1_22
- b_1_0·b_5_13 + c_4_9·a_1_32 + c_4_8·a_1_32 + c_4_8·a_1_22
- b_1_1·a_5_8 + a_1_3·b_5_13 + c_4_9·a_1_3·b_1_1 + c_4_8·a_1_3·b_1_1 + c_4_8·a_1_2·a_1_3
- b_1_1·a_5_8 + a_1_2·b_5_13 + c_4_9·a_1_2·b_1_1 + c_4_8·a_1_2·b_1_1 + c_4_8·a_1_22
- a_5_82
- b_5_122 + b_1_05·b_5_12 + c_4_8·b_1_06 + c_4_9·a_1_3·b_1_05 + c_4_92·b_1_12
+ c_4_92·b_1_02 + c_4_92·a_1_32 + c_4_92·a_1_22 + c_4_82·a_1_22
- a_5_8·b_5_13 + a_5_8·b_5_12 + a_1_3·b_1_04·b_5_12 + c_4_9·a_1_3·b_5_12
+ c_4_8·a_1_3·b_1_05 + c_4_8·a_1_2·b_5_13 + c_4_8·a_1_2·b_1_15 + c_4_92·a_1_3·b_1_1 + c_4_92·a_1_3·b_1_0 + c_4_8·c_4_9·a_1_2·b_1_1 + c_4_82·a_1_2·b_1_1 + c_4_92·a_1_32 + c_4_92·a_1_2·a_1_3 + c_4_8·c_4_9·a_1_2·a_1_3 + c_4_82·a_1_22
- a_5_8·b_5_12 + a_1_3·b_1_04·b_5_12 + c_4_9·a_1_3·b_5_12 + c_4_9·a_1_2·b_5_13
+ c_4_8·a_1_3·b_1_05 + c_4_92·a_1_3·b_1_1 + c_4_92·a_1_3·b_1_0 + c_4_92·a_1_2·b_1_1 + c_4_8·c_4_9·a_1_2·b_1_1 + c_4_92·a_1_32 + c_4_92·a_1_2·a_1_3 + c_4_8·c_4_9·a_1_2·a_1_3 + c_4_8·c_4_9·a_1_22
- b_5_12·b_5_13 + a_5_8·b_5_13 + a_5_8·b_5_12 + a_1_3·b_1_04·b_5_12 + c_4_9·b_1_1·b_5_13
+ c_4_9·a_1_3·b_5_12 + c_4_8·a_1_3·b_1_05 + c_4_8·a_1_2·b_1_15 + c_4_92·a_1_3·b_1_0 + c_4_92·a_1_2·b_1_1 + c_4_8·c_4_9·a_1_3·b_1_1 + c_4_82·a_1_2·b_1_1 + c_4_92·a_1_2·a_1_3 + c_4_8·c_4_9·a_1_32 + c_4_82·a_1_22
- b_5_132 + c_4_8·b_1_16 + c_4_92·b_1_12 + c_4_82·b_1_12 + c_4_82·a_1_22
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 + b_1_02, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 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_2 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_0 → 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
- a_5_8 → 0, an element of degree 5
- b_5_12 → 0, an element of degree 5
- b_5_13 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_2 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_0 → c_1_2, an element of degree 1
- b_1_1 → 0, 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_1·c_1_23 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- a_5_8 → 0, an element of degree 5
- b_5_12 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
- b_5_13 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_2 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_0 → 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
- a_5_8 → 0, an element of degree 5
- b_5_12 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_0·c_1_24 + c_1_04·c_1_2, an element of degree 5
- b_5_13 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
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