Simon King
David J. Green
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Cohomology of group number 1322 of order 128
General information on the group
- The group has 4 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 3.
- 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 3.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t8 + 3·t7 + 4·t6 + 5·t5 + 8·t4 + 5·t3 + 4·t2 + 3·t + 1) |
| (t + 1)2 · (t − 1)3 · (t2 + 1)3 |
- The a-invariants are -∞,-∞,-∞,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 11 minimal generators of maximal degree 4:
- 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_1_3, a nilpotent element of degree 1
- a_3_5, a nilpotent element of degree 3
- a_3_6, a nilpotent element of degree 3
- a_3_7, a nilpotent element of degree 3
- a_3_8, a nilpotent element of degree 3
- c_4_13, a Duflot regular element of degree 4
- c_4_14, a Duflot regular element of degree 4
- c_4_15, a Duflot regular element of degree 4
Ring relations
There are 26 minimal relations of maximal degree 9:
- a_1_22 + a_1_1·a_1_2 + a_1_12
- a_1_12 + a_1_0·a_1_3 + a_1_0·a_1_2 + a_1_0·a_1_1 + a_1_02
- a_1_32 + a_1_12 + a_1_0·a_1_2
- a_1_0·a_1_1·a_1_2 + a_1_02·a_1_2 + a_1_03
- a_1_0·a_1_1·a_1_3 + a_1_0·a_1_1·a_1_2 + a_1_02·a_1_1 + a_1_03
- a_1_0·a_1_1·a_1_2 + a_1_02·a_1_3 + a_1_02·a_1_1 + a_1_03
- a_1_3·a_3_6 + a_1_3·a_3_5 + a_1_2·a_3_5 + a_1_1·a_3_8 + a_1_1·a_3_6 + a_1_0·a_3_7
+ a_1_0·a_3_6 + a_1_0·a_3_5
- a_1_3·a_3_5 + a_1_2·a_3_5 + a_1_1·a_3_6 + a_1_0·a_3_8 + a_1_0·a_3_7 + a_1_0·a_3_5
- a_1_3·a_3_8 + a_1_0·a_3_6
- a_1_0·a_1_1·a_3_6 + a_1_02·a_3_6
- a_1_2·a_1_3·a_3_7 + a_1_1·a_1_3·a_3_7 + a_1_1·a_1_2·a_3_5 + a_1_0·a_1_3·a_3_7
+ a_1_0·a_1_2·a_3_6 + a_1_0·a_1_1·a_3_8 + a_1_02·a_3_5
- a_1_0·a_1_1·a_3_6 + a_1_0·a_1_1·a_3_5 + a_1_02·a_3_8 + a_1_02·a_3_7 + a_1_02·a_3_5
- a_1_2·a_1_3·a_3_7 + a_1_1·a_1_2·a_3_8 + a_1_1·a_1_2·a_3_7 + a_1_0·a_1_2·a_3_5
+ a_1_0·a_1_1·a_3_7 + a_1_02·a_3_5
- a_1_0·a_1_3·a_3_7 + a_1_0·a_1_2·a_3_8 + a_1_0·a_1_2·a_3_7 + a_1_0·a_1_2·a_3_6
+ a_1_0·a_1_1·a_3_7 + a_1_02·a_3_7 + a_1_02·a_3_5
- a_3_52 + a_1_03·a_3_5 + c_4_14·a_1_02 + c_4_13·a_1_1·a_1_2 + c_4_13·a_1_0·a_1_3
+ c_4_13·a_1_0·a_1_1 + c_4_13·a_1_02
- a_3_82 + a_3_6·a_3_8 + a_1_02·a_1_2·a_3_5 + c_4_15·a_1_1·a_1_3 + c_4_15·a_1_02
+ c_4_13·a_1_0·a_1_3 + c_4_13·a_1_02
- a_3_82 + a_1_02·a_1_1·a_3_7 + a_1_03·a_3_5 + c_4_15·a_1_0·a_1_3 + c_4_15·a_1_0·a_1_1
+ c_4_13·a_1_02
- a_3_72 + a_3_62 + a_3_52 + a_1_02·a_1_2·a_3_5 + a_1_02·a_1_1·a_3_7 + a_1_03·a_3_7
+ a_1_03·a_3_5 + c_4_15·a_1_1·a_1_2 + c_4_15·a_1_02 + c_4_14·a_1_1·a_1_2 + c_4_14·a_1_0·a_1_2 + c_4_13·a_1_1·a_1_2 + c_4_13·a_1_0·a_1_2 + c_4_13·a_1_02
- a_3_82 + a_3_62 + a_1_02·a_1_2·a_3_5 + a_1_03·a_3_7 + c_4_15·a_1_0·a_1_2
+ c_4_13·a_1_0·a_1_3 + c_4_13·a_1_0·a_1_1
- a_1_0·a_3_5·a_3_8 + a_1_0·a_3_5·a_3_7 + a_1_0·a_3_5·a_3_6 + c_4_14·a_1_02·a_1_1
+ c_4_14·a_1_03 + c_4_13·a_1_02·a_1_1
- a_1_1·a_3_5·a_3_8 + a_1_1·a_3_5·a_3_7 + a_1_0·a_3_6·a_3_7 + a_1_0·a_3_5·a_3_8
+ a_1_0·a_3_5·a_3_7 + c_4_15·a_1_03 + c_4_14·a_1_02·a_1_1 + c_4_13·a_1_1·a_1_2·a_1_3 + c_4_13·a_1_02·a_1_2
- a_1_2·a_3_7·a_3_8 + a_1_2·a_3_6·a_3_7 + a_1_2·a_3_5·a_3_7 + c_4_15·a_1_1·a_1_2·a_1_3
+ c_4_14·a_1_0·a_1_2·a_1_3 + c_4_13·a_1_02·a_1_2 + c_4_13·a_1_03
- a_1_2·a_3_5·a_3_7 + a_1_1·a_3_6·a_3_7 + a_1_0·a_3_5·a_3_8 + a_1_0·a_3_5·a_3_6
+ c_4_15·a_1_0·a_1_2·a_1_3 + c_4_14·a_1_0·a_1_2·a_1_3 + c_4_13·a_1_1·a_1_2·a_1_3 + c_4_13·a_1_0·a_1_2·a_1_3
- a_1_2·a_3_7·a_3_8 + a_1_2·a_3_5·a_3_7 + a_1_1·a_3_7·a_3_8 + a_1_1·a_3_6·a_3_7
+ a_1_1·a_3_5·a_3_7 + a_1_0·a_3_5·a_3_7 + a_1_0·a_3_5·a_3_6 + c_4_15·a_1_02·a_1_2 + c_4_14·a_1_0·a_1_2·a_1_3 + c_4_14·a_1_02·a_1_1 + c_4_13·a_1_02·a_1_2 + c_4_13·a_1_02·a_1_1 + c_4_13·a_1_03
- a_3_5·a_3_7·a_3_8 + a_3_5·a_3_6·a_3_7 + a_1_03·a_3_5·a_3_7 + c_4_15·a_1_1·a_1_2·a_3_5
+ c_4_15·a_1_0·a_1_2·a_3_7 + c_4_15·a_1_0·a_1_2·a_3_6 + c_4_15·a_1_0·a_1_1·a_3_7 + c_4_15·a_1_02·a_3_5 + c_4_14·a_1_0·a_1_2·a_3_8 + c_4_14·a_1_0·a_1_2·a_3_6 + c_4_14·a_1_0·a_1_1·a_3_8 + c_4_14·a_1_02·a_3_6 + c_4_13·a_1_1·a_1_2·a_3_5 + c_4_13·a_1_0·a_1_2·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_8 + c_4_13·a_1_0·a_1_1·a_3_7
- a_3_5·a_3_7·a_3_8 + c_4_15·a_1_0·a_1_2·a_3_7 + c_4_15·a_1_0·a_1_2·a_3_6
+ c_4_15·a_1_0·a_1_2·a_3_5 + c_4_15·a_1_0·a_1_1·a_3_7 + c_4_15·a_1_02·a_3_5 + c_4_14·a_1_0·a_1_2·a_3_8 + c_4_14·a_1_02·a_3_8 + c_4_14·a_1_02·a_3_6 + c_4_13·a_1_1·a_1_2·a_3_8 + c_4_13·a_1_0·a_1_2·a_3_8 + c_4_13·a_1_0·a_1_2·a_3_5 + c_4_13·a_1_0·a_1_1·a_3_8 + c_4_13·a_1_02·a_3_7 + c_4_13·a_1_02·a_3_6
Data used for Benson′s test
- Benson′s completion test succeeded in degree 9.
- 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_13, a Duflot regular element of degree 4
- c_4_14, a Duflot regular element of degree 4
- c_4_15, a Duflot regular element of degree 4
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 9].
- 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 3
- 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_1_3 → 0, an element of degree 1
- a_3_5 → 0, an element of degree 3
- a_3_6 → 0, an element of degree 3
- a_3_7 → 0, an element of degree 3
- a_3_8 → 0, an element of degree 3
- c_4_13 → c_1_04, an element of degree 4
- c_4_14 → c_1_14, an element of degree 4
- c_4_15 → c_1_24 + c_1_14 + c_1_04, an element of degree 4
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