Simon King
David J. Green
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Cohomology of group number 2201 of order 128
General information on the group
- The group has 5 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 5.
- Its center has rank 3.
- It has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 4 and 5, respectively.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 5 and depth 3.
- The depth coincides with the Duflot bound.
- The Poincaré series is
t5 − 2·t4 + t3 − t − 1 |
| (t + 1) · (t − 1)5 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-∞,-6,-5,-5. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 11 minimal generators of maximal degree 6:
- 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
- b_1_3, an element of degree 1
- c_1_4, a Duflot regular element of degree 1
- b_3_23, an element of degree 3
- b_4_37, an element of degree 4
- b_4_38, an element of degree 4
- c_4_39, a Duflot regular element of degree 4
- c_4_40, a Duflot regular element of degree 4
- b_6_108, an element of degree 6
Ring relations
There are 22 minimal relations of maximal degree 12:
- b_1_1·b_1_2 + b_1_12 + a_1_0·b_1_2
- b_1_1·b_1_3 + a_1_0·b_1_1 + a_1_02
- a_1_02·b_1_2
- a_1_02·b_1_3 + a_1_02·b_1_1 + a_1_03
- a_1_02·b_3_23
- a_1_0·b_1_2·b_3_23 + a_1_0·b_1_1·b_3_23 + b_4_37·a_1_0
- b_4_37·b_1_1 + a_1_0·b_1_2·b_3_23 + a_1_0·b_1_1·b_3_23
- b_1_2·b_1_3·b_3_23 + b_1_22·b_3_23 + b_1_12·b_3_23 + b_4_38·b_1_3 + b_4_37·b_1_2
- a_1_0·b_1_1·b_3_23 + b_4_38·a_1_0
- b_1_12·b_3_23 + b_4_38·b_1_1 + a_1_0·b_1_14
- b_3_232 + b_1_23·b_3_23 + b_1_16 + b_4_37·b_1_32 + b_4_37·b_1_2·b_1_3
+ a_1_0·b_1_15 + b_4_37·a_1_0·b_1_2 + c_4_40·b_1_32 + c_4_40·b_1_12 + c_4_39·b_1_22 + c_4_40·a_1_02
- b_1_34·b_3_23 + b_6_108·b_1_3 + b_4_37·b_3_23 + b_4_37·b_1_23 + b_4_38·a_1_0·b_1_12
+ b_4_37·a_1_0·b_1_22 + c_4_40·b_1_33 + c_4_40·b_1_22·b_1_3 + c_4_39·b_1_2·b_1_32 + c_4_39·b_1_22·b_1_3 + c_4_39·b_1_23 + c_4_39·b_1_13 + c_4_40·a_1_0·b_1_32 + c_4_39·a_1_0·b_1_32 + c_4_39·a_1_0·b_1_12 + c_4_40·a_1_02·b_1_1 + c_4_39·a_1_02·b_1_1 + c_4_40·a_1_03 + c_4_39·a_1_03
- b_1_24·b_3_23 + b_1_17 + b_6_108·b_1_2 + b_4_38·b_3_23 + b_4_38·b_1_33
+ b_4_38·b_1_2·b_1_32 + b_4_38·b_1_22·b_1_3 + b_4_38·b_1_23 + b_4_37·b_1_22·b_1_3 + a_1_0·b_1_16 + b_4_37·a_1_0·b_1_22 + c_4_40·b_1_22·b_1_3 + c_4_40·b_1_23 + c_4_40·b_1_13 + c_4_39·b_1_22·b_1_3 + c_4_40·a_1_0·b_1_12 + c_4_39·a_1_02·b_1_1 + c_4_40·a_1_03
- a_1_0·b_1_33·b_3_23 + b_6_108·a_1_0 + b_4_38·a_1_0·b_1_12 + b_4_37·a_1_0·b_1_22
+ c_4_40·a_1_0·b_1_32 + c_4_40·a_1_0·b_1_22 + c_4_39·a_1_0·b_1_12 + c_4_40·a_1_03
- b_6_108·b_1_1 + b_4_38·b_1_13 + b_4_38·a_1_0·b_1_12 + b_4_37·a_1_0·b_1_22
+ c_4_40·b_1_13 + c_4_39·b_1_13 + c_4_40·a_1_0·b_1_22 + c_4_40·a_1_0·b_1_12 + c_4_40·a_1_02·b_1_1 + c_4_39·a_1_03
- b_4_38·b_1_2·b_3_23 + b_4_38·b_1_22·b_1_32 + b_4_38·b_1_23·b_1_3 + b_4_382
+ b_4_37·b_1_2·b_3_23 + b_4_37·b_1_23·b_1_3 + b_4_37·b_1_24 + b_4_37·b_4_38 + b_4_38·a_1_0·b_3_23 + b_4_37·a_1_0·b_1_23 + c_4_39·b_1_23·b_1_3 + c_4_39·b_1_24 + c_4_39·b_1_14 + c_4_39·a_1_0·b_1_13
- b_4_38·b_1_2·b_1_33 + b_4_38·b_1_23·b_1_3 + b_4_38·b_1_1·b_3_23 + b_4_382
+ b_4_37·b_1_22·b_1_32 + b_4_37·b_1_24 + b_4_372 + b_4_37·a_1_0·b_1_23 + c_4_39·b_1_22·b_1_32 + c_4_39·b_1_24 + c_4_39·b_1_14
- b_6_108·b_1_22 + b_4_38·b_1_2·b_1_33 + b_4_38·b_1_23·b_1_3 + b_4_38·b_1_24
+ b_4_37·b_1_2·b_3_23 + b_4_37·b_1_22·b_1_32 + b_4_37·b_1_23·b_1_3 + b_4_37·b_4_38 + c_4_40·b_1_22·b_1_32 + c_4_40·b_1_23·b_1_3 + c_4_40·b_1_24 + c_4_39·b_1_24 + c_4_40·a_1_0·b_1_13
- b_6_108·b_3_23 + b_4_38·b_1_24·b_1_3 + b_4_382·b_1_3 + b_4_382·b_1_2
+ b_4_37·b_1_35 + b_4_37·b_1_2·b_1_34 + b_4_37·b_1_22·b_1_33 + b_4_37·b_1_23·b_1_32 + b_4_37·b_1_25 + b_4_372·b_1_3 + b_4_382·a_1_0 + c_4_40·b_1_32·b_3_23 + c_4_40·b_1_35 + c_4_40·b_1_22·b_3_23 + c_4_40·b_1_22·b_1_33 + c_4_39·b_1_22·b_3_23 + c_4_39·b_1_22·b_1_33 + c_4_39·b_1_23·b_1_32 + c_4_39·b_1_25 + c_4_39·b_1_15 + b_4_38·c_4_39·b_1_3 + b_4_38·c_4_39·b_1_2 + b_4_38·c_4_39·b_1_1 + b_4_37·c_4_40·b_1_3 + b_4_37·c_4_39·b_1_2 + c_4_40·a_1_0·b_1_3·b_3_23 + c_4_39·a_1_0·b_1_3·b_3_23 + c_4_39·a_1_0·b_1_24
- b_4_38·b_6_108 + b_4_382·b_1_32 + b_4_382·b_1_2·b_1_3 + b_4_382·b_1_12
+ b_4_37·b_1_2·b_1_35 + b_4_37·b_1_22·b_1_34 + b_4_37·b_1_23·b_3_23 + b_4_37·b_1_24·b_1_32 + b_4_37·b_1_25·b_1_3 + b_4_37·b_4_38·b_1_32 + b_4_37·b_4_38·b_1_2·b_1_3 + b_4_37·b_4_38·b_1_22 + b_4_372·b_1_22 + c_4_40·b_1_2·b_1_35 + c_4_40·b_1_23·b_3_23 + c_4_40·b_1_24·b_1_32 + c_4_39·b_1_23·b_1_33 + c_4_39·b_1_24·b_1_32 + c_4_39·b_1_25·b_1_3 + b_4_38·c_4_40·b_1_32 + b_4_38·c_4_40·b_1_22 + b_4_38·c_4_40·b_1_12 + b_4_38·c_4_39·b_1_2·b_1_3 + b_4_38·c_4_39·b_1_12 + b_4_37·c_4_40·b_1_2·b_1_3 + b_4_37·c_4_40·b_1_22 + c_4_40·a_1_0·b_1_15 + c_4_39·a_1_0·b_1_15 + b_4_38·c_4_40·a_1_0·b_1_1 + b_4_38·c_4_39·a_1_0·b_1_1
- b_4_37·b_1_33·b_3_23 + b_4_37·b_1_22·b_1_34 + b_4_37·b_1_24·b_1_32
+ b_4_37·b_6_108 + b_4_37·b_4_38·b_1_2·b_1_3 + b_4_372·b_1_22 + b_4_372·a_1_0·b_1_2 + c_4_40·b_1_22·b_1_34 + c_4_40·b_1_23·b_3_23 + c_4_40·b_1_23·b_1_33 + c_4_39·b_1_24·b_1_32 + b_4_38·c_4_40·b_1_2·b_1_3 + b_4_38·c_4_40·b_1_12 + b_4_38·c_4_39·b_1_2·b_1_3 + b_4_38·c_4_39·b_1_22 + b_4_38·c_4_39·b_1_12 + b_4_37·c_4_40·b_1_32 + b_4_37·c_4_39·b_1_2·b_1_3 + c_4_40·a_1_0·b_1_15 + b_4_38·c_4_39·a_1_0·b_1_1
- b_6_1082 + b_4_382·b_1_23·b_1_3 + b_4_382·b_1_24 + b_4_37·b_1_38
+ b_4_37·b_1_2·b_1_37 + b_4_37·b_1_25·b_1_33 + b_4_37·b_1_26·b_1_32 + b_4_37·b_4_38·b_1_23·b_1_3 + b_4_37·b_4_382 + b_4_372·b_1_2·b_3_23 + b_4_372·b_1_2·b_1_33 + b_4_372·b_1_23·b_1_3 + b_4_372·b_1_24 + b_4_373 + b_4_382·a_1_0·b_1_13 + c_4_40·b_1_38 + c_4_40·b_1_25·b_1_33 + c_4_39·b_1_22·b_1_36 + c_4_39·b_1_23·b_1_35 + c_4_39·b_1_24·b_1_34 + c_4_39·b_1_25·b_1_33 + c_4_39·b_1_26·b_1_32 + b_4_38·c_4_40·b_1_22·b_1_32 + b_4_38·c_4_40·b_1_23·b_1_3 + b_4_38·c_4_39·b_1_22·b_1_32 + b_4_38·c_4_39·b_1_23·b_1_3 + b_4_38·c_4_39·b_1_1·b_3_23 + b_4_382·c_4_39 + b_4_37·c_4_40·b_1_22·b_1_32 + b_4_37·c_4_40·b_1_24 + b_4_37·c_4_39·b_1_22·b_1_32 + b_4_37·c_4_39·b_1_23·b_1_3 + b_4_372·c_4_40 + b_4_38·c_4_40·a_1_0·b_1_13 + b_4_37·c_4_40·a_1_0·b_1_23 + c_4_402·b_1_34 + c_4_402·b_1_24 + c_4_39·c_4_40·b_1_22·b_1_32 + c_4_39·c_4_40·b_1_24 + c_4_39·c_4_40·b_1_14 + c_4_392·b_1_22·b_1_32 + c_4_392·b_1_14
Data used for Benson′s test
- Benson′s completion test succeeded in degree 12.
- 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_1_4, a Duflot regular element of degree 1
- c_4_39, a Duflot regular element of degree 4
- c_4_40, a Duflot regular element of degree 4
- b_1_32 + b_1_2·b_1_3 + b_1_22, an element of degree 2
- b_1_32, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 3, 6, 8].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is 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
- b_1_3 → 0, an element of degree 1
- c_1_4 → c_1_0, an element of degree 1
- b_3_23 → 0, an element of degree 3
- b_4_37 → 0, an element of degree 4
- b_4_38 → 0, an element of degree 4
- c_4_39 → c_1_14, an element of degree 4
- c_4_40 → c_1_24, an element of degree 4
- b_6_108 → 0, an element of degree 6
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_3, 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_1_4 → c_1_0, an element of degree 1
- b_3_23 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
- b_4_37 → 0, an element of degree 4
- b_4_38 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_1·c_1_33 + c_1_12·c_1_32, an element of degree 4
- c_4_39 → c_1_12·c_1_32 + c_1_14, an element of degree 4
- c_4_40 → c_1_34 + c_1_2·c_1_33 + c_1_24 + c_1_1·c_1_33 + c_1_12·c_1_32, an element of degree 4
- b_6_108 → c_1_36 + c_1_22·c_1_34 + c_1_24·c_1_32 + c_1_12·c_1_34 + c_1_14·c_1_32, an element of degree 6
Restriction map to a maximal el. ab. subgp. of rank 5
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- b_1_3 → c_1_4, an element of degree 1
- c_1_4 → c_1_0, an element of degree 1
- b_3_23 → c_1_2·c_1_3·c_1_4 + c_1_22·c_1_4 + c_1_1·c_1_3·c_1_4 + c_1_12·c_1_3, an element of degree 3
- b_4_37 → c_1_2·c_1_32·c_1_4 + c_1_22·c_1_3·c_1_4 + c_1_1·c_1_3·c_1_42 + c_1_1·c_1_32·c_1_4
+ c_1_12·c_1_3·c_1_4 + c_1_12·c_1_32, an element of degree 4
- b_4_38 → c_1_2·c_1_32·c_1_4 + c_1_22·c_1_3·c_1_4, an element of degree 4
- c_4_39 → c_1_2·c_1_3·c_1_42 + c_1_2·c_1_32·c_1_4 + c_1_22·c_1_42 + c_1_22·c_1_3·c_1_4
+ c_1_1·c_1_3·c_1_42 + c_1_1·c_1_32·c_1_4 + c_1_12·c_1_42 + c_1_12·c_1_3·c_1_4 + c_1_12·c_1_32 + c_1_14, an element of degree 4
- c_4_40 → c_1_2·c_1_32·c_1_4 + c_1_22·c_1_3·c_1_4 + c_1_22·c_1_32 + c_1_24
+ c_1_1·c_1_3·c_1_42 + c_1_12·c_1_3·c_1_4, an element of degree 4
- b_6_108 → c_1_2·c_1_3·c_1_44 + c_1_2·c_1_34·c_1_4 + c_1_22·c_1_44
+ c_1_22·c_1_32·c_1_42 + c_1_22·c_1_34 + c_1_24·c_1_42 + c_1_24·c_1_3·c_1_4 + c_1_24·c_1_32 + c_1_1·c_1_32·c_1_43 + c_1_1·c_1_33·c_1_42 + c_1_1·c_1_34·c_1_4 + c_1_1·c_1_2·c_1_32·c_1_42 + c_1_1·c_1_22·c_1_3·c_1_42 + c_1_12·c_1_3·c_1_43 + c_1_12·c_1_32·c_1_42 + c_1_12·c_1_33·c_1_4 + c_1_12·c_1_34 + c_1_12·c_1_2·c_1_32·c_1_4 + c_1_12·c_1_22·c_1_3·c_1_4 + c_1_14·c_1_3·c_1_4, an element of degree 6
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