Simon King
David J. Green
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Singular
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Cohomology of group number 525 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 3 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 3.
- The depth exceeds the Duflot bound, which is 2.
- The Poincaré series is
( − 1) · (t2 + 1)2 |
| (t − 1)3 · (t4 + 1) |
- The a-invariants are -∞,-∞,-∞,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 12 minimal generators of maximal degree 8:
- a_1_0, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- a_2_2, a nilpotent element of degree 2
- b_2_3, an element of degree 2
- b_2_4, an element of degree 2
- b_2_7, an element of degree 2
- c_2_6, a Duflot regular element of degree 2
- a_3_6, a nilpotent element of degree 3
- a_3_14, a nilpotent element of degree 3
- a_5_35, a nilpotent element of degree 5
- c_8_88, a Duflot regular element of degree 8
Ring relations
There are 36 minimal relations of maximal degree 10:
- a_1_02
- a_1_0·b_1_1
- a_1_0·b_1_2
- a_2_2·a_1_0
- b_2_3·a_1_0
- b_2_4·a_1_0
- b_2_4·b_1_1 + b_2_3·b_1_2
- b_1_23 + b_1_1·b_1_22 + b_2_7·b_1_1
- b_2_32 + c_2_6·b_1_12
- b_2_3·b_2_4 + c_2_6·b_1_1·b_1_2
- b_2_42 + c_2_6·b_1_22
- b_2_7·b_1_1·b_1_2 + b_2_7·b_1_12 + a_2_2·b_1_22 + a_2_2·b_1_1·b_1_2 + a_2_22
- b_2_7·b_1_22 + b_2_7·b_1_1·b_1_2 + b_1_2·a_3_6 + a_2_2·b_2_4
- a_1_0·a_3_6
- b_1_1·a_3_6 + a_2_2·b_1_22 + a_2_2·b_1_1·b_1_2 + a_2_2·b_2_3 + a_2_22
- b_1_2·a_3_14 + a_2_2·b_1_22 + a_2_2·b_2_7 + a_2_2·b_2_4
- b_2_4·b_1_22 + b_2_3·b_1_22 + b_2_3·b_2_7 + a_1_0·a_3_14
- b_1_1·a_3_14 + a_2_2·b_1_22 + a_2_2·b_2_3
- b_1_22·a_3_6 + a_2_2·b_2_7·b_1_1 + a_2_2·b_2_4·b_1_2 + a_2_22·b_1_2
- b_2_3·b_2_7·b_1_2 + b_2_3·b_2_7·b_1_1 + b_2_3·a_3_6 + a_2_2·c_2_6·b_1_1
- b_2_4·b_2_7·b_1_2 + b_2_3·b_2_7·b_1_2 + b_2_4·a_3_6 + a_2_2·c_2_6·b_1_2
- b_2_3·b_2_7·b_1_2 + b_2_3·b_2_7·b_1_1 + a_2_2·b_2_7·b_1_2 + a_2_2·b_2_7·b_1_1
+ a_2_2·b_2_4·b_1_2 + a_2_2·b_2_3·b_1_2 + a_2_2·a_3_6
- b_2_3·a_3_14 + a_2_2·b_2_4·b_1_2 + a_2_2·c_2_6·b_1_1
- b_2_72·b_1_2 + b_2_7·a_3_6 + b_2_4·a_3_14 + a_2_2·b_2_7·b_1_1 + a_2_2·b_2_4·b_1_2
+ a_2_22·b_1_2 + a_2_2·c_2_6·b_1_2
- b_2_3·b_2_7·b_1_2 + b_2_3·b_2_7·b_1_1 + a_2_2·b_2_7·b_1_2 + a_2_2·b_2_7·b_1_1
+ a_2_2·b_2_4·b_1_2 + a_2_2·b_2_3·b_1_2 + a_2_2·a_3_14 + a_2_22·b_1_2
- b_2_7·b_1_2·a_3_6 + a_2_2·b_2_4·b_2_7 + a_3_62 + a_2_22·c_2_6
- b_2_4·b_1_2·a_3_6 + a_2_2·b_2_72 + a_2_2·b_2_3·b_2_7 + a_3_6·a_3_14
+ a_2_2·c_2_6·b_1_22 + a_2_22·c_2_6
- b_2_7·b_1_2·a_3_6 + b_2_4·b_1_2·a_3_6 + a_2_2·b_2_72 + a_2_2·b_2_4·b_2_7
+ a_2_2·b_2_3·b_2_7 + a_3_142 + b_2_7·a_1_0·a_3_14 + a_2_22·b_1_22 + a_2_22·b_2_7 + a_2_22·b_2_4 + a_2_2·c_2_6·b_1_22 + a_2_22·c_2_6
- b_1_2·a_5_35 + b_2_7·b_1_2·a_3_6 + a_2_2·b_2_7·b_1_12 + a_2_2·b_2_3·b_1_22
+ a_2_2·b_2_3·b_1_1·b_1_2 + a_2_2·b_2_3·b_2_7 + a_2_22·b_1_1·b_1_2 + b_2_7·c_2_6·b_1_12 + c_2_6·b_1_2·a_3_6 + a_2_2·c_2_6·b_1_1·b_1_2 + a_2_2·b_2_4·c_2_6 + a_2_22·c_2_6
- b_2_4·b_1_2·a_3_6 + a_2_2·b_2_3·b_2_7 + a_1_0·a_5_35 + a_2_22·b_2_4
+ a_2_2·c_2_6·b_1_22
- b_1_1·a_5_35 + a_2_2·b_1_12·b_1_22 + a_2_2·b_1_13·b_1_2 + a_2_2·b_2_3·b_1_22
+ a_2_2·b_2_3·b_1_12 + a_2_2·b_2_3·b_2_7 + a_2_22·b_1_22 + a_2_22·b_1_1·b_1_2 + a_2_22·b_1_12 + a_2_22·b_2_7 + b_2_7·c_2_6·b_1_12 + a_2_2·c_2_6·b_1_22 + a_2_22·c_2_6
- b_2_3·a_5_35 + a_2_2·b_2_3·b_1_1·b_1_22 + a_2_2·b_2_3·b_1_12·b_1_2
+ a_2_2·b_2_4·a_3_6 + a_2_22·b_2_4·b_1_2 + a_2_22·b_2_3·b_1_2 + a_2_22·b_2_3·b_1_1 + b_2_3·b_2_7·c_2_6·b_1_1 + a_2_2·c_2_6·b_1_1·b_1_22 + a_2_2·c_2_6·b_1_13 + a_2_2·b_2_7·c_2_6·b_1_2 + a_2_2·b_2_4·c_2_6·b_1_2 + a_2_2·c_2_6·a_3_6 + a_2_22·c_2_6·b_1_2
- b_2_4·a_5_35 + b_2_4·b_2_7·a_3_6 + a_2_2·b_2_3·b_2_7·b_1_1 + a_2_22·b_2_3·b_1_2
+ b_2_3·b_2_7·c_2_6·b_1_1 + b_2_4·c_2_6·a_3_6 + a_2_2·c_2_6·b_1_1·b_1_22 + a_2_2·c_2_6·b_1_12·b_1_2 + a_2_2·b_2_7·c_2_6·b_1_2 + a_2_2·b_2_3·c_2_6·b_1_2 + a_2_2·c_2_6·a_3_6 + a_2_2·c_2_62·b_1_2
- a_2_2·a_5_35 + a_2_2·b_2_7·a_3_6 + a_2_22·b_1_1·b_1_22 + a_2_22·b_1_12·b_1_2
+ a_2_22·b_2_4·b_1_2 + a_2_22·b_2_3·b_1_1 + a_2_2·b_2_7·c_2_6·b_1_2 + a_2_22·c_2_6·b_1_2
- a_3_6·a_5_35 + b_2_7·a_3_62 + a_2_22·b_2_3·b_2_7 + a_2_2·b_2_4·b_2_7·c_2_6
+ c_2_6·a_3_62 + a_2_22·c_2_6·b_1_22 + a_2_22·c_2_6·b_1_12 + a_2_22·b_2_7·c_2_6 + a_2_22·b_2_4·c_2_6 + a_2_22·c_2_62
- a_5_352 + b_2_72·a_3_62 + a_2_22·c_2_6·b_1_12·b_1_22 + a_2_22·c_2_6·b_1_14
+ a_2_2·b_2_7·c_2_62·b_1_12 + c_2_62·a_3_62 + a_2_22·c_2_62·b_1_22 + a_2_22·c_2_62·b_1_1·b_1_2 + a_2_22·c_2_63
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_6, a Duflot regular element of degree 2
- c_8_88, a Duflot regular element of degree 8
- b_1_12 + b_2_7, an element of degree 2
- 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 2
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- a_2_2 → 0, an element of degree 2
- b_2_3 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- b_2_7 → 0, an element of degree 2
- c_2_6 → c_1_02, an element of degree 2
- a_3_6 → 0, an element of degree 3
- a_3_14 → 0, an element of degree 3
- a_5_35 → 0, an element of degree 5
- c_8_88 → c_1_18, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → 0, an element of degree 1
- a_2_2 → 0, an element of degree 2
- b_2_3 → c_1_0·c_1_2, an element of degree 2
- b_2_4 → 0, an element of degree 2
- b_2_7 → 0, an element of degree 2
- c_2_6 → c_1_02, an element of degree 2
- a_3_6 → 0, an element of degree 3
- a_3_14 → 0, an element of degree 3
- a_5_35 → 0, an element of degree 5
- c_8_88 → c_1_14·c_1_24 + c_1_18, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- a_2_2 → 0, an element of degree 2
- b_2_3 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- b_2_7 → c_1_22, an element of degree 2
- c_2_6 → c_1_02, an element of degree 2
- a_3_6 → 0, an element of degree 3
- a_3_14 → 0, an element of degree 3
- a_5_35 → 0, an element of degree 5
- c_8_88 → c_1_14·c_1_24 + c_1_18 + c_1_04·c_1_24, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- a_2_2 → 0, an element of degree 2
- b_2_3 → c_1_0·c_1_2, an element of degree 2
- b_2_4 → c_1_0·c_1_2, an element of degree 2
- b_2_7 → 0, an element of degree 2
- c_2_6 → c_1_02, an element of degree 2
- a_3_6 → 0, an element of degree 3
- a_3_14 → 0, an element of degree 3
- a_5_35 → 0, an element of degree 5
- c_8_88 → c_1_28 + c_1_14·c_1_24 + c_1_18 + c_1_02·c_1_26 + c_1_04·c_1_24
+ c_1_06·c_1_22, an element of degree 8
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