Simon King
David J. Green
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Cohomology of group number 1450 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 4.
- Its center has rank 3.
- It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 3.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t5 − 2·t4 − t3 − t2 − 2·t − 1) |
| (t + 1)2 · (t − 1)4 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-∞,-4,-4. 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_1, a nilpotent element of degree 1
- a_1_0, a nilpotent element of degree 1
- b_1_2, an element of degree 1
- b_1_3, an element of degree 1
- c_2_7, a Duflot regular element of degree 2
- b_3_10, an element of degree 3
- b_3_11, an element of degree 3
- b_3_12, an element of degree 3
- c_4_20, a Duflot regular element of degree 4
- c_4_21, a Duflot regular element of degree 4
Ring relations
There are 15 minimal relations of maximal degree 7:
- a_1_1·b_1_2
- a_1_0·b_1_2 + a_1_1·b_1_3 + a_1_02 + a_1_1·a_1_0
- b_1_2·b_1_3 + a_1_0·b_1_3 + a_1_0·b_1_2 + a_1_12
- a_1_03 + a_1_1·a_1_02 + a_1_12·a_1_0
- a_1_02·b_1_3 + a_1_1·a_1_02 + a_1_13
- a_1_1·b_3_10 + c_2_7·a_1_1·b_1_3 + c_2_7·a_1_1·a_1_0
- b_1_3·b_3_11 + b_1_3·b_3_10 + b_1_2·b_3_12 + b_1_2·b_3_11 + a_1_1·b_3_12 + c_2_7·b_1_22
+ c_2_7·a_1_1·a_1_0
- b_1_2·b_3_12 + b_1_2·b_3_11 + a_1_0·b_3_12 + a_1_0·b_3_10 + a_1_1·b_3_11 + c_2_7·b_1_22
+ c_2_7·a_1_02 + c_2_7·a_1_12
- a_1_02·b_3_11 + a_1_1·a_1_0·b_3_11 + a_1_12·b_3_12 + c_2_7·a_1_1·a_1_02
+ c_2_7·a_1_12·a_1_0 + c_2_7·a_1_13
- a_1_02·b_3_12 + a_1_02·b_3_11 + a_1_12·b_3_11 + c_2_7·a_1_12·a_1_0
- b_3_102 + b_1_23·b_3_11 + b_1_23·b_3_10 + b_1_26 + a_1_0·b_1_32·b_3_10
+ c_4_21·b_1_22 + c_2_7·b_1_34 + c_2_7·a_1_0·b_1_33 + c_2_72·b_1_32 + c_2_72·b_1_22 + c_2_72·a_1_02
- b_3_112 + b_3_102 + b_1_23·b_3_11 + b_1_23·b_3_10 + b_1_26 + c_4_20·b_1_22
+ c_2_7·b_1_24 + c_4_21·a_1_02 + c_4_21·a_1_12 + c_4_20·a_1_02 + c_2_72·a_1_12
- b_3_11·b_3_12 + b_3_10·b_3_12 + b_3_10·b_3_11 + b_3_102 + b_1_23·b_3_11
+ b_1_23·b_3_10 + b_1_26 + a_1_13·b_3_11 + c_4_20·b_1_22 + c_2_7·b_1_2·b_3_11 + c_2_7·b_1_2·b_3_10 + c_2_7·b_1_24 + c_4_21·a_1_0·b_1_3 + c_4_21·a_1_1·b_1_3 + c_4_20·a_1_1·b_1_3 + c_2_7·a_1_0·b_1_33 + c_2_7·a_1_1·b_3_12 + c_4_21·a_1_1·a_1_0 + c_4_21·a_1_12 + c_4_20·a_1_02 + c_2_72·a_1_1·b_1_3 + c_2_72·a_1_1·a_1_0
- b_3_122 + b_3_102 + b_1_23·b_3_11 + b_1_23·b_3_10 + b_1_26 + a_1_13·b_3_11
+ c_4_21·b_1_32 + c_4_20·b_1_22 + c_2_7·b_1_34 + c_2_7·b_1_24 + c_4_21·a_1_12 + c_4_20·a_1_12 + c_2_72·b_1_22
- a_1_0·b_3_10·b_3_11 + c_2_7·a_1_0·b_1_34 + c_2_7·a_1_02·b_3_10
+ c_2_7·a_1_1·a_1_0·b_3_11 + c_2_72·a_1_0·b_1_32 + c_2_72·a_1_1·a_1_02 + c_2_72·a_1_12·a_1_0 + c_2_72·a_1_13
Data used for Benson′s test
- Benson′s completion test succeeded in degree 8.
- However, the last relation was already found in degree 7 and the last generator in degree 4.
- The following is a filter regular homogeneous system of parameters:
- c_2_7, a Duflot regular element of degree 2
- c_4_20, a Duflot regular element of degree 4
- c_4_21, a Duflot regular element of degree 4
- b_1_32 + b_1_22, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 6, 8].
- 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 3
- a_1_1 → 0, an element of degree 1
- a_1_0 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- c_2_7 → c_1_22, an element of degree 2
- b_3_10 → 0, an element of degree 3
- b_3_11 → 0, an element of degree 3
- b_3_12 → 0, an element of degree 3
- c_4_20 → c_1_24 + c_1_14, an element of degree 4
- c_4_21 → c_1_14 + c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_1 → 0, an element of degree 1
- a_1_0 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → c_1_3, an element of degree 1
- c_2_7 → c_1_22, an element of degree 2
- b_3_10 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
- b_3_11 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
- b_3_12 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32
+ c_1_02·c_1_3, an element of degree 3
- c_4_20 → c_1_2·c_1_33 + c_1_24 + c_1_12·c_1_32 + c_1_14, an element of degree 4
- c_4_21 → c_1_22·c_1_32 + c_1_12·c_1_32 + c_1_14 + c_1_02·c_1_32 + c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_1 → 0, an element of degree 1
- a_1_0 → 0, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- b_1_3 → 0, an element of degree 1
- c_2_7 → c_1_2·c_1_3 + c_1_22, an element of degree 2
- b_3_10 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32
+ c_1_02·c_1_3, an element of degree 3
- b_3_11 → c_1_33 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
- b_3_12 → c_1_33 + c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
- c_4_20 → c_1_34 + c_1_22·c_1_32 + c_1_24 + c_1_1·c_1_33 + c_1_14, an element of degree 4
- c_4_21 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_1·c_1_33 + c_1_14 + c_1_02·c_1_32
+ c_1_04, an element of degree 4
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