Simon King
David J. Green
Cohomology
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Cohomology of group number 360 of order 128
General information on the group
- The group has 3 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
- The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 10 minimal generators of maximal degree 5:
- a_1_1, a nilpotent element of degree 1
- b_1_0, an element of degree 1
- b_1_2, an element of degree 1
- a_2_3, a nilpotent element of degree 2
- b_3_5, an element of degree 3
- b_3_6, an element of degree 3
- b_4_7, an element of degree 4
- c_4_8, a Duflot regular element of degree 4
- c_4_9, a Duflot regular element of degree 4
- b_5_13, an element of degree 5
Ring relations
There are 27 minimal relations of maximal degree 10:
- a_1_1·b_1_0
- b_1_0·b_1_2
- a_1_1·b_1_22
- a_1_12·b_1_2 + a_1_13
- a_2_3·b_1_0
- a_2_3·a_1_1·b_1_2 + a_2_32 + a_2_3·a_1_12
- b_1_2·b_3_5 + b_1_24 + a_2_3·b_1_22 + a_2_3·a_1_1·b_1_2 + a_2_3·a_1_12
- a_1_1·b_3_5 + a_2_3·a_1_1·b_1_2
- b_1_2·b_3_6 + a_2_3·a_1_12
- a_1_1·b_3_6 + a_2_3·a_1_1·b_1_2 + a_2_3·a_1_12
- a_2_3·b_3_5 + a_2_3·b_1_23 + a_2_3·a_1_13
- a_2_3·b_3_6 + a_2_3·a_1_13
- b_4_7·a_1_1 + a_2_3·a_1_13
- b_1_02·b_3_6 + b_1_02·b_3_5 + b_4_7·b_1_0 + a_2_3·a_1_13
- b_3_62 + c_4_9·b_1_02 + c_4_8·b_1_02
- b_3_52 + b_1_26 + b_4_7·b_1_02 + c_4_8·b_1_02
- b_1_2·b_5_13 + b_1_26 + a_2_3·b_1_24 + a_2_3·b_4_7 + c_4_9·b_1_22 + c_4_8·b_1_22
+ c_4_9·a_1_1·b_1_2
- a_1_1·b_5_13 + c_4_9·a_1_1·b_1_2 + c_4_8·a_1_1·b_1_2 + c_4_9·a_1_12
- b_3_5·b_3_6 + b_1_0·b_5_13 + b_1_03·b_3_5 + c_4_8·b_1_02
- b_4_7·b_3_6 + b_4_7·b_3_5 + b_4_7·b_1_23 + b_4_7·b_1_03 + a_2_3·b_4_7·b_1_2
+ c_4_9·b_1_03
- b_1_02·b_5_13 + b_1_04·b_3_5 + b_4_7·b_3_5 + b_4_7·b_1_23 + b_4_7·b_1_03
+ a_2_3·b_4_7·b_1_2
- a_2_3·b_5_13 + a_2_3·b_1_25 + a_2_3·c_4_9·b_1_2 + a_2_3·c_4_8·b_1_2
+ a_2_3·c_4_9·a_1_1
- b_4_7·b_1_04 + b_4_72 + a_2_3·b_1_26 + c_4_9·b_1_04 + c_4_8·b_1_24
- b_3_5·b_5_13 + b_1_28 + b_4_7·b_1_0·b_3_5 + a_2_3·b_4_7·b_1_22 + c_4_9·b_1_24
+ c_4_9·b_1_04 + c_4_8·b_1_24 + c_4_8·b_1_0·b_3_6 + c_4_8·b_1_0·b_3_5 + c_4_8·b_1_04 + a_2_3·c_4_9·b_1_22 + a_2_3·c_4_8·b_1_22 + a_2_32·c_4_8 + a_2_3·c_4_9·a_1_12
- b_3_6·b_5_13 + b_4_7·b_1_0·b_3_5 + b_4_7·b_1_04 + c_4_9·b_1_0·b_3_5
+ c_4_8·b_1_0·b_3_6 + c_4_8·b_1_0·b_3_5 + c_4_8·b_1_04 + a_2_32·c_4_9 + a_2_3·c_4_9·a_1_12 + a_2_3·c_4_8·a_1_12
- b_4_7·b_5_13 + b_4_7·b_1_25 + b_4_72·b_1_0 + a_2_3·b_4_7·b_1_23
+ c_4_9·b_1_02·b_3_5 + b_4_7·c_4_9·b_1_2 + b_4_7·c_4_8·b_1_2 + a_2_3·c_4_8·b_1_23 + a_2_3·c_4_9·a_1_13
- b_5_132 + b_1_210 + b_4_72·b_1_02 + c_4_9·b_1_06 + c_4_8·b_1_06
+ b_4_7·c_4_9·b_1_02 + b_4_7·c_4_8·b_1_02 + c_4_92·b_1_22 + c_4_8·c_4_9·b_1_02 + c_4_82·b_1_22 + c_4_92·a_1_12
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_22 + b_1_02, 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_1 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- a_2_3 → 0, an element of degree 2
- b_3_5 → 0, an element of degree 3
- b_3_6 → 0, an element of degree 3
- b_4_7 → 0, an element of degree 4
- c_4_8 → c_1_14 + c_1_04, an element of degree 4
- c_4_9 → c_1_04, an element of degree 4
- b_5_13 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_1 → 0, an element of degree 1
- b_1_0 → c_1_2, an element of degree 1
- b_1_2 → 0, an element of degree 1
- a_2_3 → 0, an element of degree 2
- b_3_5 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- b_3_6 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
- b_4_7 → c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
- c_4_8 → c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_23 + c_1_04, an element of degree 4
- c_4_9 → c_1_0·c_1_23 + c_1_04, an element of degree 4
- b_5_13 → c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_0·c_1_1·c_1_23 + c_1_0·c_1_12·c_1_22
+ c_1_02·c_1_23 + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_2, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_1 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- a_2_3 → 0, an element of degree 2
- b_3_5 → c_1_23, an element of degree 3
- b_3_6 → 0, an element of degree 3
- b_4_7 → c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
- c_4_8 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_9 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- b_5_13 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
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