Simon King
David J. Green
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Cohomology of group number 1234 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 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 3.
- The depth coincides with the Duflot bound.
- The Poincaré series is
(2) · (t3 + 1/2·t + 1/2) |
| (t + 1) · (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 12 minimal generators of maximal degree 5:
- a_1_0, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- b_1_3, an element of degree 1
- c_2_7, a Duflot regular element of degree 2
- a_3_9, a nilpotent element of degree 3
- a_3_12, a nilpotent element of degree 3
- b_3_10, an element of degree 3
- b_3_13, an element of degree 3
- c_4_22, a Duflot regular element of degree 4
- c_4_23, a Duflot regular element of degree 4
- b_5_37, an element of degree 5
Ring relations
There are 28 minimal relations of maximal degree 10:
- a_1_22 + a_1_0·a_1_2
- a_1_12 + a_1_0·a_1_1 + a_1_02
- a_1_2·b_1_3 + a_1_2·a_1_1 + a_1_02
- a_1_03
- a_1_02·b_1_3 + a_1_02·a_1_1
- a_1_2·a_3_9 + a_1_0·a_3_12 + c_2_7·a_1_2·a_1_1 + c_2_7·a_1_0·a_1_1
- a_1_2·a_3_12 + a_1_2·a_3_9
- b_1_3·a_3_9 + a_1_0·b_3_10 + a_1_1·a_3_9 + c_2_7·a_1_1·b_1_3 + c_2_7·a_1_0·b_1_3
+ c_2_7·a_1_02
- a_1_2·b_3_10 + a_1_0·a_3_9 + c_2_7·a_1_0·a_1_1 + c_2_7·a_1_02
- b_1_3·a_3_12 + b_1_3·a_3_9 + a_1_0·b_3_13 + a_1_1·a_3_12 + a_1_1·a_3_9 + a_1_2·a_3_9
+ c_2_7·a_1_2·a_1_1
- a_1_2·b_3_13 + a_1_0·a_3_9 + c_2_7·a_1_0·a_1_1
- a_1_02·b_3_10
- a_3_122 + a_3_9·a_3_12 + c_2_7·a_1_1·a_3_12 + c_2_7·a_1_1·a_3_9
- a_3_12·b_3_13 + a_3_9·b_3_13 + a_3_9·b_3_10 + a_3_122 + c_2_7·a_1_1·b_3_10
+ c_2_7·a_1_0·b_3_10 + c_2_7·a_1_0·b_1_33 + c_2_7·a_1_0·a_1_1·b_1_32 + c_2_72·a_1_0·b_1_3 + c_2_72·a_1_02
- a_3_12·b_3_10 + a_3_9·b_3_13 + a_3_9·b_3_10 + a_3_122 + c_2_7·a_1_1·b_3_13
+ c_2_7·a_1_0·b_3_13 + c_2_7·a_1_0·a_3_12 + c_2_72·a_1_02
- b_3_132 + b_3_102 + a_1_0·b_1_32·b_3_13 + a_1_0·b_1_32·b_3_10 + a_3_122
+ a_1_02·a_1_1·a_3_9 + c_2_7·b_1_34 + c_2_7·a_1_0·b_1_33 + c_4_22·a_1_02 + c_2_7·a_1_0·a_1_1·b_1_32 + c_2_72·b_1_32
- a_3_122 + c_4_22·a_1_0·a_1_2 + c_2_72·a_1_0·a_1_1 + c_2_72·a_1_02
- b_3_132 + a_1_0·b_1_32·b_3_13 + a_1_0·b_1_32·b_3_10 + a_3_122
+ a_1_0·a_1_1·b_1_3·b_3_13 + a_1_0·a_1_1·b_1_3·b_3_10 + c_4_23·b_1_32 + c_4_22·b_1_32 + c_2_7·a_1_1·b_1_33 + c_2_7·a_1_0·b_1_33 + c_4_23·a_1_0·a_1_1 + c_4_22·a_1_0·a_1_1 + c_2_7·a_1_0·a_1_1·b_1_32 + c_2_72·a_1_0·a_1_1 + c_2_72·a_1_02
- b_3_102 + a_3_9·b_3_10 + a_3_92 + a_1_0·a_1_1·b_1_3·b_3_13
+ a_1_0·a_1_1·b_1_3·b_3_10 + a_1_02·a_1_1·a_3_9 + c_4_23·b_1_32 + c_4_22·b_1_32 + c_2_7·b_1_34 + c_4_23·a_1_0·b_1_3 + c_4_22·a_1_0·b_1_3 + c_2_7·a_1_1·b_3_10 + c_2_7·a_1_1·b_1_33 + c_2_7·a_1_0·b_3_10 + c_2_7·a_1_0·b_1_33 + c_4_23·a_1_02 + c_2_72·b_1_32 + c_2_72·a_1_0·b_1_3 + c_2_72·a_1_0·a_1_1
- b_3_132 + a_3_9·b_3_10 + a_1_0·b_1_32·b_3_13 + a_1_0·b_1_32·b_3_10 + a_3_122
+ a_1_0·a_1_1·b_1_3·b_3_13 + a_1_0·a_1_1·b_1_3·b_3_10 + c_4_23·b_1_32 + c_4_22·b_1_32 + c_4_23·a_1_0·b_1_3 + c_4_22·a_1_0·b_1_3 + c_2_7·a_1_1·b_3_10 + c_2_7·a_1_1·b_1_33 + c_2_7·a_1_0·b_3_10 + c_4_23·a_1_0·a_1_2 + c_2_7·a_1_0·a_1_1·b_1_32 + c_2_72·a_1_0·b_1_3 + c_2_72·a_1_02
- b_3_10·b_3_13 + b_3_102 + b_1_3·b_5_37 + b_1_33·b_3_13 + a_3_9·b_3_13 + a_1_1·b_5_37
+ a_1_1·b_1_32·b_3_13 + a_1_1·b_1_32·b_3_10 + a_3_122 + a_3_92 + a_1_0·a_1_1·b_1_3·b_3_10 + a_1_02·a_1_1·a_3_9 + c_4_23·b_1_32 + c_2_7·b_1_3·b_3_13 + c_2_7·b_1_3·b_3_10 + c_4_23·a_1_1·b_1_3 + c_4_22·a_1_1·b_1_3 + c_2_7·a_1_1·b_3_10 + c_2_7·a_1_0·b_3_10 + c_2_7·a_1_0·b_1_33 + c_4_22·a_1_0·a_1_1 + c_2_72·a_1_0·b_1_3 + c_2_72·a_1_0·a_1_1
- a_3_9·b_3_13 + a_3_9·b_3_10 + a_1_0·b_5_37 + a_1_0·b_1_32·b_3_13 + a_3_122 + a_3_92
+ a_1_0·a_1_1·b_1_3·b_3_10 + c_4_23·a_1_0·b_1_3 + c_2_7·a_1_1·b_3_13 + c_2_7·a_1_1·b_3_10 + c_4_22·a_1_0·a_1_1 + c_2_7·a_1_0·a_3_9 + c_2_72·a_1_0·a_1_1
- b_3_102 + a_3_9·b_3_10 + a_1_2·b_5_37 + a_3_92 + a_1_0·a_1_1·b_1_3·b_3_13
+ a_1_0·a_1_1·b_1_3·b_3_10 + a_1_02·a_1_1·a_3_9 + c_4_23·b_1_32 + c_4_22·b_1_32 + c_2_7·b_1_34 + c_4_23·a_1_0·b_1_3 + c_4_22·a_1_0·b_1_3 + c_2_7·a_1_1·b_3_10 + c_2_7·a_1_1·b_1_33 + c_2_7·a_1_0·b_3_10 + c_2_7·a_1_0·b_1_33 + c_4_23·a_1_2·a_1_1 + c_4_22·a_1_2·a_1_1 + c_2_7·a_1_0·a_3_12 + c_2_7·a_1_0·a_3_9 + c_2_72·b_1_32 + c_2_72·a_1_0·b_1_3 + c_2_72·a_1_0·a_1_1
- b_3_13·b_5_37 + b_3_10·b_5_37 + b_1_33·b_5_37 + b_1_35·b_3_13 + a_1_1·b_1_34·b_3_10
+ a_1_0·b_1_34·b_3_13 + a_1_0·b_1_34·b_3_10 + c_4_23·b_1_3·b_3_13 + c_4_23·b_1_3·b_3_10 + c_4_23·b_1_34 + c_2_7·b_1_33·b_3_13 + c_2_7·b_1_36 + c_4_23·a_1_0·b_3_13 + c_4_23·a_1_0·b_3_10 + c_4_23·a_1_0·b_1_33 + c_4_22·a_1_1·b_3_13 + c_4_22·a_1_1·b_3_10 + c_4_22·a_1_1·b_1_33 + c_4_22·a_1_0·b_3_13 + c_4_22·a_1_0·b_3_10 + c_4_22·a_1_0·b_1_33 + c_2_7·a_1_1·b_1_32·b_3_10 + c_2_7·a_1_0·b_1_35 + c_4_22·a_1_0·a_3_12 + c_4_22·a_1_0·a_3_9 + c_2_7·a_1_0·a_1_1·b_1_3·b_3_13 + c_2_7·a_1_0·a_1_1·b_1_34 + c_2_72·b_1_3·b_3_10 + c_2_72·a_1_1·b_3_10 + c_2_72·a_1_0·b_3_10 + c_2_7·c_4_23·a_1_02 + c_2_7·c_4_22·a_1_0·a_1_2 + c_2_72·a_1_0·a_1_1·b_1_32 + c_2_73·b_1_32 + c_2_73·a_1_0·b_1_3 + c_2_73·a_1_02
- b_3_13·b_5_37 + a_3_9·b_5_37 + a_1_1·b_1_32·b_5_37 + a_1_1·b_1_34·b_3_13
+ a_1_0·b_1_34·b_3_13 + a_1_0·b_1_34·b_3_10 + a_1_0·a_1_1·b_1_33·b_3_13 + c_4_23·b_1_3·b_3_10 + c_4_23·b_1_34 + c_4_22·b_1_3·b_3_13 + c_4_22·b_1_3·b_3_10 + c_4_22·b_1_34 + c_2_7·b_1_3·b_5_37 + c_4_23·a_1_1·b_3_13 + c_4_23·a_1_1·b_3_10 + c_4_23·a_1_0·b_3_13 + c_4_23·a_1_0·b_3_10 + c_4_23·a_1_0·b_1_33 + c_4_22·a_1_1·b_3_10 + c_4_22·a_1_1·b_1_33 + c_4_22·a_1_0·b_3_13 + c_4_22·a_1_0·b_1_33 + c_2_7·a_1_1·b_1_32·b_3_10 + c_2_7·a_1_0·b_1_32·b_3_10 + c_4_23·a_1_1·a_3_9 + c_4_23·a_1_0·a_3_12 + c_4_22·a_1_1·a_3_9 + c_4_22·a_1_0·a_3_12 + c_4_22·a_1_0·a_1_1·b_1_32 + c_2_7·a_1_0·a_1_1·b_1_3·b_3_13 + c_2_7·c_4_23·b_1_32 + c_2_72·b_1_3·b_3_10 + c_2_72·b_1_34 + c_2_7·c_4_23·a_1_1·b_1_3 + c_2_7·c_4_22·a_1_1·b_1_3 + c_2_7·c_4_22·a_1_0·b_1_3 + c_2_72·a_1_1·b_3_10 + c_2_72·a_1_1·b_1_33 + c_2_72·a_1_0·b_3_10 + c_2_72·a_1_0·b_1_33 + c_2_7·c_4_23·a_1_0·a_1_2 + c_2_7·c_4_22·a_1_0·a_1_2 + c_2_7·c_4_22·a_1_02 + c_2_72·a_1_0·a_1_1·b_1_32 + c_2_73·b_1_32 + c_2_73·a_1_0·b_1_3 + c_2_73·a_1_02
- a_3_9·b_5_37 + a_1_0·b_1_32·b_5_37 + a_1_0·b_1_34·b_3_13
+ a_1_0·a_1_1·b_1_33·b_3_10 + c_4_23·a_1_0·b_3_13 + c_4_22·a_1_0·b_3_13 + c_4_22·a_1_0·b_3_10 + c_4_22·a_1_0·b_1_33 + c_2_7·a_1_1·b_5_37 + c_2_7·a_1_0·b_1_32·b_3_13 + c_2_7·a_1_0·b_1_35 + c_4_23·a_1_1·a_3_9 + c_4_23·a_1_0·a_3_12 + c_4_23·a_1_0·a_3_9 + c_4_22·a_1_1·a_3_9 + c_4_22·a_1_0·a_3_12 + c_4_22·a_1_0·a_3_9 + c_4_22·a_1_0·a_1_1·b_1_32 + c_2_7·a_1_0·a_1_1·b_1_3·b_3_13 + c_2_7·a_1_0·a_1_1·b_1_34 + c_2_7·c_4_23·a_1_0·b_1_3 + c_2_72·a_1_0·b_1_33 + c_2_7·c_4_22·a_1_0·a_1_1
- a_3_12·b_5_37 + a_3_9·b_5_37 + a_1_0·a_1_1·b_1_3·b_5_37 + a_1_0·a_1_1·b_1_33·b_3_13
+ c_4_23·a_1_0·b_3_10 + c_4_23·a_1_0·b_1_33 + c_4_22·a_1_0·b_3_13 + c_4_22·a_1_0·b_3_10 + c_4_22·a_1_0·b_1_33 + c_2_7·a_1_0·b_5_37 + c_4_23·a_1_1·a_3_12 + c_4_23·a_1_1·a_3_9 + c_4_23·a_1_0·a_3_9 + c_4_23·a_1_0·a_1_1·b_1_32 + c_4_22·a_1_1·a_3_12 + c_4_22·a_1_1·a_3_9 + c_4_22·a_1_0·a_3_12 + c_4_22·a_1_0·a_3_9 + c_2_7·a_1_0·a_1_1·b_1_3·b_3_10 + c_2_7·a_1_02·a_1_1·a_3_9 + c_2_7·c_4_23·a_1_0·b_1_3 + c_2_72·a_1_0·b_3_10 + c_2_72·a_1_0·b_1_33 + c_2_7·c_4_23·a_1_0·a_1_1 + c_2_7·c_4_23·a_1_0·a_1_2 + c_2_7·c_4_22·a_1_0·a_1_1 + c_2_7·c_4_22·a_1_0·a_1_2 + c_2_72·a_1_0·a_3_9 + c_2_73·a_1_0·b_1_3
- b_5_372 + a_1_0·b_1_36·b_3_13 + a_1_0·b_1_36·b_3_10 + a_1_0·a_1_1·b_1_35·b_3_13
+ a_1_0·a_1_1·b_1_35·b_3_10 + c_4_23·b_1_36 + c_4_22·b_1_36 + c_4_23·a_1_0·b_1_32·b_3_13 + c_4_23·a_1_0·b_1_32·b_3_10 + c_4_22·a_1_0·b_1_32·b_3_13 + c_4_22·a_1_0·b_1_32·b_3_10 + c_2_7·a_1_1·b_1_37 + c_2_7·a_1_0·b_1_34·b_3_13 + c_2_7·a_1_0·b_1_34·b_3_10 + c_2_7·a_1_0·b_1_37 + c_4_23·a_1_02·a_1_1·a_3_9 + c_4_22·a_1_02·a_1_1·a_3_9 + c_4_232·b_1_32 + c_2_7·c_4_23·b_1_34 + c_2_7·c_4_22·b_1_34 + c_2_72·b_1_36 + c_2_7·c_4_23·a_1_0·b_1_33 + c_2_7·c_4_22·a_1_0·b_1_33 + c_2_72·a_1_1·b_1_35 + c_2_72·a_1_0·b_1_35 + c_4_232·a_1_0·a_1_2 + c_4_232·a_1_02 + c_4_22·c_4_23·a_1_0·a_1_2 + c_4_22·c_4_23·a_1_02 + c_4_222·a_1_0·a_1_1 + c_4_222·a_1_02 + c_2_7·c_4_23·a_1_0·a_1_1·b_1_32 + c_2_7·c_4_22·a_1_0·a_1_1·b_1_32 + c_2_72·a_1_0·a_1_1·b_1_3·b_3_13 + c_2_72·a_1_0·a_1_1·b_1_3·b_3_10 + c_2_72·a_1_02·a_1_1·a_3_9 + c_2_72·c_4_23·b_1_32 + c_2_72·c_4_22·b_1_32 + c_2_73·b_1_34 + c_2_73·a_1_1·b_1_33 + c_2_72·c_4_23·a_1_0·a_1_1 + c_2_72·c_4_23·a_1_0·a_1_2 + c_2_72·c_4_23·a_1_02 + c_2_72·c_4_22·a_1_0·a_1_1
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_7, a Duflot regular element of degree 2
- c_4_22, a Duflot regular element of degree 4
- c_4_23, a Duflot regular element of degree 4
- b_1_32, 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_0 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- c_2_7 → c_1_12, an element of degree 2
- a_3_9 → 0, an element of degree 3
- a_3_12 → 0, an element of degree 3
- b_3_10 → 0, an element of degree 3
- b_3_13 → 0, an element of degree 3
- c_4_22 → c_1_24, an element of degree 4
- c_4_23 → c_1_24 + c_1_04, an element of degree 4
- b_5_37 → 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
- a_1_2 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_1_3 → c_1_3, an element of degree 1
- c_2_7 → c_1_12, an element of degree 2
- a_3_9 → 0, an element of degree 3
- a_3_12 → 0, an element of degree 3
- b_3_10 → c_1_33 + c_1_12·c_1_3 + c_1_02·c_1_3, an element of degree 3
- b_3_13 → c_1_33 + c_1_1·c_1_32 + c_1_02·c_1_3, an element of degree 3
- c_4_22 → c_1_34 + c_1_22·c_1_32 + c_1_24 + c_1_12·c_1_32 + c_1_02·c_1_32, an element of degree 4
- c_4_23 → c_1_22·c_1_32 + c_1_24 + c_1_02·c_1_32 + c_1_04, an element of degree 4
- b_5_37 → c_1_35 + c_1_22·c_1_33 + c_1_24·c_1_3 + c_1_12·c_1_33 + c_1_02·c_1_1·c_1_32
+ c_1_02·c_1_12·c_1_3 + c_1_04·c_1_3, an element of degree 5
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