Simon King
David J. Green
Cohomology
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Singular
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Cohomology of group number 77 of order 128
General information on the group
- The group has 2 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 2.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
t2 − t + 1 |
| (t + 1) · (t − 1)4 · (t2 + 1) |
- The a-invariants are -∞,-∞,-5,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 11 minimal generators of maximal degree 5:
- a_1_0, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- a_2_1, a nilpotent element of degree 2
- b_2_2, an element of degree 2
- c_2_3, a Duflot regular element of degree 2
- a_3_3, a nilpotent element of degree 3
- b_3_4, an element of degree 3
- b_3_6, an element of degree 3
- b_4_7, an element of degree 4
- c_4_11, a Duflot regular element of degree 4
- b_5_16, an element of degree 5
Ring relations
There are 33 minimal relations of maximal degree 10:
- a_1_02
- a_1_0·b_1_1
- a_2_1·a_1_0
- a_2_1·b_1_1
- b_2_2·a_1_0
- a_2_12
- a_1_0·a_3_3
- b_1_1·a_3_3
- a_1_0·b_3_4
- a_1_0·b_3_6
- a_2_1·a_3_3
- a_2_1·b_3_4
- a_2_1·b_3_6
- b_4_7·a_1_0
- b_1_12·b_3_6 + b_4_7·b_1_1 + b_2_2·b_3_4 + c_2_3·b_1_13
- a_2_1·b_2_22 + a_3_32
- a_3_3·b_3_4
- a_3_3·b_3_6 + a_2_1·b_2_22
- b_3_62 + b_3_42 + b_2_2·b_1_1·b_3_6 + b_2_2·b_1_1·b_3_4 + b_2_22·b_1_12
+ c_4_11·b_1_12 + b_2_2·c_2_3·b_1_12
- a_2_1·b_4_7
- b_3_42 + b_1_13·b_3_4 + b_4_7·b_1_12 + b_2_2·b_1_1·b_3_4 + b_2_2·b_1_14
- a_1_0·b_5_16
- b_3_4·b_3_6 + b_3_42 + b_1_1·b_5_16 + b_2_2·b_1_1·b_3_4 + c_2_3·b_1_14
+ b_2_2·c_2_3·b_1_12 + c_2_32·b_1_12
- b_4_7·a_3_3
- b_1_14·b_3_4 + b_4_7·b_3_6 + b_4_7·b_1_13 + b_2_2·b_5_16 + b_2_2·b_1_12·b_3_4
+ b_2_2·b_1_15 + b_2_22·b_3_4 + c_4_11·b_1_13 + c_2_3·b_4_7·b_1_1 + b_2_2·c_2_3·b_3_4 + b_2_2·c_2_3·b_1_13 + b_2_22·c_2_3·b_1_1 + b_2_2·c_2_3·a_3_3 + c_2_32·b_1_13 + b_2_2·c_2_32·b_1_1
- a_2_1·b_5_16
- b_1_12·b_5_16 + b_1_14·b_3_4 + b_4_7·b_3_4 + b_4_7·b_1_13 + b_2_2·b_1_12·b_3_4
+ b_2_2·b_1_15 + b_2_2·b_4_7·b_1_1 + b_2_22·b_3_4 + b_2_22·b_1_13 + c_2_3·b_1_12·b_3_4 + c_2_3·b_1_15 + b_2_2·c_2_3·b_1_13 + c_2_32·b_1_13
- a_3_3·b_5_16 + c_2_3·a_3_32
- b_4_7·b_1_1·b_3_6 + b_4_72 + b_2_2·b_1_1·b_5_16 + b_2_2·b_1_13·b_3_4
+ b_2_2·b_4_7·b_1_12 + b_2_22·b_1_1·b_3_6 + b_2_22·b_1_1·b_3_4 + b_2_22·b_1_14 + b_2_23·b_1_12 + b_2_2·a_3_32 + c_2_3·b_4_7·b_1_12 + b_2_2·c_2_3·b_1_1·b_3_4 + b_2_2·c_2_3·b_1_14 + b_2_2·c_2_32·b_1_12
- b_3_4·b_5_16 + b_2_2·b_1_13·b_3_4 + b_2_2·b_4_7·b_1_12 + b_2_22·b_1_1·b_3_4
+ c_4_11·b_1_14 + c_2_3·b_4_7·b_1_12 + b_2_2·c_2_3·b_1_14 + c_2_32·b_1_1·b_3_4 + c_2_32·b_1_14
- b_3_6·b_5_16 + b_2_2·b_1_13·b_3_4 + b_2_2·b_4_7·b_1_12 + c_4_11·b_1_1·b_3_4
+ c_4_11·b_1_14 + c_2_3·b_1_13·b_3_4 + b_2_2·c_2_3·b_1_1·b_3_6 + b_2_2·c_2_3·b_1_1·b_3_4 + b_2_2·c_2_3·b_1_14 + c_2_3·a_3_32 + c_2_32·b_1_1·b_3_6
- b_4_7·b_5_16 + b_2_2·b_4_7·b_3_6 + b_2_22·b_5_16 + b_2_22·b_1_15
+ b_2_22·b_4_7·b_1_1 + c_4_11·b_1_12·b_3_4 + c_4_11·b_1_15 + c_2_3·b_4_7·b_3_4 + c_2_3·b_4_7·b_1_13 + b_2_22·c_2_3·b_3_4 + b_2_22·c_2_3·b_1_13 + b_2_23·c_2_3·b_1_1 + b_2_22·c_2_3·a_3_3 + c_2_32·b_1_12·b_3_4 + c_2_32·b_1_15 + c_2_32·b_4_7·b_1_1 + b_2_22·c_2_32·b_1_1
- b_5_162 + b_2_23·b_1_14 + c_4_11·b_1_13·b_3_4 + b_4_7·c_4_11·b_1_12
+ b_2_2·c_4_11·b_1_1·b_3_4 + c_2_32·b_1_16 + b_2_2·c_2_32·b_1_14 + b_2_22·c_2_32·b_1_12 + c_2_32·a_3_32 + c_2_34·b_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_2_3, a Duflot regular element of degree 2
- c_4_11, a Duflot regular element of degree 4
- b_1_1·b_3_6 + b_1_1·b_3_4 + b_1_14 + b_2_2·b_1_12 + b_2_22 + c_2_3·b_1_12, an element of degree 4
- b_3_4 + b_2_2·b_1_1, an element of degree 3
- The Raw Filter Degree Type of that HSOP is [-1, -1, 1, 6, 9].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
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
- b_1_1 → 0, an element of degree 1
- a_2_1 → 0, an element of degree 2
- b_2_2 → 0, an element of degree 2
- c_2_3 → c_1_02, an element of degree 2
- a_3_3 → 0, an element of degree 3
- b_3_4 → 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_11 → c_1_14 + c_1_04, an element of degree 4
- b_5_16 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- a_2_1 → 0, an element of degree 2
- b_2_2 → c_1_32 + c_1_2·c_1_3 + c_1_1·c_1_2, an element of degree 2
- c_2_3 → c_1_32 + c_1_2·c_1_3 + c_1_0·c_1_2 + c_1_02, an element of degree 2
- a_3_3 → 0, an element of degree 3
- b_3_4 → c_1_1·c_1_22, an element of degree 3
- b_3_6 → c_1_12·c_1_2 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- b_4_7 → c_1_22·c_1_32 + c_1_23·c_1_3 + 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_11 → c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_13·c_1_2 + c_1_14
+ c_1_02·c_1_22 + c_1_04, an element of degree 4
- b_5_16 → c_1_23·c_1_32 + c_1_24·c_1_3 + c_1_13·c_1_22 + c_1_0·c_1_22·c_1_32
+ c_1_0·c_1_23·c_1_3 + c_1_0·c_1_24 + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3 + c_1_04·c_1_2, an element of degree 5
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