Simon King
David J. Green
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Cohomology of group number 1961 of order 128
General information on the group
- The group has 4 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 2.
- 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 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t2 + t + 1) |
| (t − 1)3 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-3,-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 8:
- 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
- b_1_3, an element of degree 1
- a_3_10, a nilpotent element of degree 3
- b_4_12, an element of degree 4
- c_4_13, a Duflot regular element of degree 4
- a_5_13, a nilpotent element of degree 5
- a_5_19, a nilpotent element of degree 5
- a_7_31, a nilpotent element of degree 7
- c_8_38, a Duflot regular element of degree 8
Ring relations
There are 28 minimal relations of maximal degree 14:
- a_1_0·a_1_2
- a_1_0·b_1_3 + a_1_22 + a_1_1·a_1_2 + a_1_12 + a_1_0·a_1_1
- a_1_03
- a_1_22·b_1_3 + a_1_1·a_1_2·b_1_3 + a_1_12·b_1_3 + a_1_13 + a_1_0·a_1_12
- a_1_0·a_3_10
- a_1_13·b_1_32 + a_1_14·b_1_3
- b_4_12·a_1_0 + a_1_1·a_1_2·a_3_10 + a_1_12·a_3_10
- b_1_32·a_3_10 + b_4_12·a_1_2 + a_1_2·b_1_3·a_3_10 + a_1_12·a_3_10
- a_3_102 + c_4_13·a_1_22
- a_3_102 + a_1_2·a_5_13 + b_4_12·a_1_1·a_1_2 + a_1_1·a_1_2·b_1_3·a_3_10
- a_1_0·a_5_19
- b_4_12·a_3_10 + c_4_13·a_1_2·b_1_32 + c_4_13·a_1_1·a_1_2·b_1_3
+ c_4_13·a_1_12·b_1_3 + c_4_13·a_1_12·a_1_2 + c_4_13·a_1_13 + c_4_13·a_1_0·a_1_12
- a_1_02·a_5_13 + b_4_12·a_1_13 + a_1_12·a_1_2·b_1_3·a_3_10
- b_1_32·a_5_13 + b_4_12·a_1_1·b_1_32 + b_4_12·a_1_1·a_1_2·b_1_3 + a_1_1·a_1_2·a_5_19
+ a_1_12·a_5_19 + b_4_12·a_1_12·a_1_2 + a_1_12·a_1_2·b_1_3·a_3_10 + c_4_13·a_1_2·b_1_32
- b_4_122 + c_4_13·b_1_34 + c_4_13·a_1_1·a_1_2·b_1_32 + c_4_13·a_1_12·b_1_32
+ c_4_13·a_1_13·b_1_3 + c_4_13·a_1_14 + c_4_13·a_1_0·a_1_13
- a_3_10·a_5_13 + c_4_13·a_1_2·a_3_10 + c_4_13·a_1_1·a_1_2·b_1_32
- a_1_0·a_7_31 + a_1_13·a_5_13 + b_4_12·a_1_14 + c_4_13·a_1_14
+ c_4_13·a_1_0·a_1_13
- a_3_10·a_5_19 + a_1_2·a_7_31 + a_1_2·b_1_32·a_5_19 + b_4_12·a_1_1·a_1_2·b_1_32
+ a_1_1·a_1_2·b_1_3·a_5_19 + b_4_12·a_1_12·a_1_2·b_1_3 + a_1_12·a_1_2·a_5_19 + c_4_13·a_1_2·a_3_10 + c_4_13·a_1_1·a_1_2·b_1_32 + c_4_13·a_1_12·a_1_2·b_1_3 + c_4_13·a_1_13·b_1_3 + c_4_13·a_1_14 + c_4_13·a_1_0·a_1_13
- b_1_32·a_7_31 + b_1_34·a_5_19 + b_4_12·a_5_19 + b_4_12·a_1_1·b_1_34
+ a_1_2·b_1_3·a_7_31 + a_1_2·b_1_33·a_5_19 + a_1_1·b_1_33·a_5_19 + b_4_12·a_1_1·a_1_2·b_1_33 + b_4_12·a_1_12·b_1_33 + a_1_12·a_7_31 + a_1_12·b_1_32·a_5_19 + b_4_12·a_1_12·a_1_2·b_1_32 + a_1_0·a_1_13·a_5_13 + c_4_13·a_1_1·b_1_34 + b_4_12·c_4_13·a_1_2 + c_4_13·a_1_12·a_1_2·b_1_32
- b_1_32·a_7_31 + b_1_34·a_5_19 + b_4_12·a_5_19 + b_4_12·a_5_13 + b_4_12·a_1_1·b_1_34
+ a_1_2·b_1_3·a_7_31 + a_1_2·b_1_33·a_5_19 + a_1_1·b_1_33·a_5_19 + b_4_12·a_1_1·a_1_2·b_1_33 + b_4_12·a_1_12·b_1_33 + a_1_1·a_1_2·a_7_31 + a_1_1·a_1_2·b_1_32·a_5_19 + a_1_12·a_1_2·b_1_3·a_5_19 + a_1_0·a_1_13·a_5_13 + c_4_13·a_1_1·a_1_2·b_1_33 + c_4_13·a_1_1·a_1_2·a_3_10 + c_4_13·a_1_12·a_3_10
- a_5_13·a_5_19 + b_4_12·a_1_1·a_5_19 + a_1_1·a_1_2·b_1_3·a_7_31
+ a_1_1·a_1_2·b_1_33·a_5_19 + b_4_12·a_1_12·a_1_2·b_1_33 + a_1_12·a_1_2·b_1_32·a_5_19 + c_4_13·a_1_2·a_5_19 + c_4_13·a_1_1·a_1_2·b_1_3·a_3_10 + c_4_13·a_1_12·a_1_2·b_1_33
- a_3_10·a_7_31 + b_4_12·a_1_2·a_5_19 + a_1_12·b_1_3·a_7_31 + a_1_12·b_1_33·a_5_19
+ c_4_13·a_1_2·a_5_19 + c_4_13·a_1_1·a_1_2·b_1_34 + b_4_12·c_4_13·a_1_1·a_1_2 + c_4_13·a_1_12·b_1_3·a_3_10 + c_4_132·a_1_22
- a_5_132 + c_8_38·a_1_02 + c_4_13·a_1_12·b_1_34 + c_4_132·a_1_22
+ c_4_132·a_1_02
- a_5_192 + a_1_2·b_1_34·a_5_19 + b_4_12·a_1_1·a_1_2·b_1_34
+ b_4_12·a_1_12·b_1_34 + a_1_1·a_1_2·b_1_33·a_5_19 + a_1_12·b_1_33·a_5_19 + a_1_12·a_1_2·a_7_31 + a_1_12·a_1_2·b_1_32·a_5_19 + c_8_38·a_1_22 + c_4_13·a_1_1·a_1_2·b_1_34 + c_4_13·a_1_12·b_1_34 + c_4_13·a_1_1·a_1_2·b_1_3·a_3_10 + c_4_13·a_1_12·b_1_3·a_3_10 + c_4_13·a_1_12·a_1_2·a_3_10 + c_4_132·a_1_22
- b_4_12·a_7_31 + b_4_12·b_1_32·a_5_19 + b_4_12·a_1_1·b_1_3·a_5_19
+ b_4_12·a_1_1·a_1_2·a_5_19 + a_1_12·a_1_2·b_1_3·a_7_31 + a_1_12·a_1_2·b_1_33·a_5_19 + c_4_13·b_1_32·a_5_19 + c_4_13·a_1_1·b_1_36 + b_4_12·c_4_13·a_1_1·b_1_32 + c_4_13·a_1_2·b_1_3·a_5_19 + c_4_13·a_1_12·b_1_35 + b_4_12·c_4_13·a_1_1·a_1_2·b_1_3 + c_4_13·a_1_12·a_5_19 + c_4_13·a_1_12·a_1_2·b_1_34 + c_4_132·a_1_2·b_1_32 + c_4_132·a_1_1·a_1_2·b_1_3 + c_4_132·a_1_12·b_1_3 + c_4_132·a_1_12·a_1_2 + c_4_132·a_1_13 + c_4_132·a_1_0·a_1_12
- a_5_13·a_7_31 + b_4_12·a_1_1·b_1_32·a_5_19 + b_4_12·a_1_1·a_1_2·b_1_3·a_5_19
+ b_4_12·a_1_12·b_1_3·a_5_19 + c_4_13·a_1_2·a_7_31 + c_4_13·a_1_1·b_1_32·a_5_19 + c_4_13·a_1_12·b_1_36 + b_4_12·c_4_13·a_1_12·b_1_32 + c_4_13·a_1_12·a_1_2·b_1_35 + c_8_38·a_1_0·a_1_13 + c_4_13·a_1_12·a_1_2·a_5_19 + c_4_13·a_1_13·a_5_13 + b_4_12·c_4_13·a_1_14 + c_4_132·a_1_1·a_1_2·b_1_32 + c_4_132·a_1_0·a_1_13
- a_5_19·a_7_31 + a_1_2·b_1_36·a_5_19 + b_4_12·a_1_2·b_1_32·a_5_19
+ b_4_12·a_1_1·b_1_32·a_5_19 + b_4_12·a_1_1·a_1_2·b_1_36 + b_4_12·a_1_12·b_1_36 + a_1_12·b_1_35·a_5_19 + b_4_12·a_1_12·b_1_3·a_5_19 + b_4_12·a_1_12·a_1_2·b_1_35 + c_8_38·a_1_2·a_3_10 + c_8_38·a_1_1·a_1_2·b_1_32 + c_8_38·a_1_12·b_1_32 + c_4_13·a_1_2·a_7_31 + c_4_13·a_1_2·b_1_32·a_5_19 + c_4_13·a_1_1·b_1_32·a_5_19 + b_4_12·c_4_13·a_1_12·b_1_32 + c_8_38·a_1_12·a_1_2·b_1_3 + c_4_13·a_1_12·a_1_2·b_1_35 + b_4_12·c_4_13·a_1_12·a_1_2·b_1_3 + b_4_12·c_4_13·a_1_14 + c_4_132·a_1_12·b_1_32
- a_7_312 + a_1_2·b_1_38·a_5_19 + b_4_12·a_1_1·a_1_2·b_1_38
+ b_4_12·a_1_12·b_1_38 + a_1_1·a_1_2·b_1_37·a_5_19 + a_1_12·b_1_37·a_5_19 + a_1_12·a_1_2·b_1_36·a_5_19 + b_4_12·a_1_12·a_1_2·b_1_32·a_5_19 + c_8_38·a_1_1·a_1_2·b_1_34 + c_8_38·a_1_12·b_1_34 + c_4_13·a_1_2·b_1_34·a_5_19 + c_4_13·a_1_1·a_1_2·b_1_38 + b_4_12·c_4_13·a_1_1·a_1_2·b_1_34 + b_4_12·c_4_13·a_1_12·b_1_34 + c_4_13·a_1_1·a_1_2·b_1_33·a_5_19 + c_4_13·a_1_12·b_1_33·a_5_19 + c_4_13·a_1_12·a_1_2·a_7_31 + c_4_13·a_1_12·a_1_2·b_1_32·a_5_19 + c_4_13·c_8_38·a_1_22 + c_4_132·a_1_12·b_1_34 + c_4_132·a_1_1·a_1_2·b_1_3·a_3_10 + c_4_132·a_1_12·b_1_3·a_3_10 + c_4_132·a_1_12·a_1_2·a_3_10
Data used for Benson′s test
- Benson′s completion test succeeded in degree 14.
- 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_8_38, a Duflot regular element of degree 8
- b_1_32, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 11].
- 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
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- a_3_10 → 0, an element of degree 3
- b_4_12 → 0, an element of degree 4
- c_4_13 → c_1_14, an element of degree 4
- a_5_13 → 0, an element of degree 5
- a_5_19 → 0, an element of degree 5
- a_7_31 → 0, an element of degree 7
- c_8_38 → c_1_18 + c_1_08, 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
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- a_3_10 → 0, an element of degree 3
- b_4_12 → c_1_12·c_1_22, an element of degree 4
- c_4_13 → c_1_14, an element of degree 4
- a_5_13 → 0, an element of degree 5
- a_5_19 → 0, an element of degree 5
- a_7_31 → 0, an element of degree 7
- c_8_38 → c_1_12·c_1_26 + c_1_14·c_1_24 + c_1_18 + c_1_04·c_1_24 + c_1_08, an element of degree 8
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