Simon King
David J. Green
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Cohomology of group number 1011 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 5.
- Its center has rank 4.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 5.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 5 and depth 4.
- The depth coincides with the Duflot bound.
- The Poincaré series is
- The a-invariants are -∞,-∞,-∞,-∞,-5,-5. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 12 minimal generators of maximal degree 4:
- a_1_0, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- c_1_3, a Duflot regular element of degree 1
- a_2_7, a nilpotent element of degree 2
- b_2_8, an element of degree 2
- c_2_9, a Duflot regular element of degree 2
- c_2_10, a Duflot regular element of degree 2
- a_3_12, a nilpotent element of degree 3
- b_3_23, an element of degree 3
- b_4_41, an element of degree 4
- c_4_45, a Duflot regular element of degree 4
Ring relations
There are 27 minimal relations of maximal degree 8:
- a_1_02
- a_1_0·a_1_2
- a_1_0·b_1_1 + a_1_22
- a_1_22·b_1_1
- a_2_7·a_1_0
- b_2_8·a_1_0 + a_2_7·a_1_2
- b_2_8·a_1_2 + a_2_7·b_1_1
- a_2_72 + c_2_9·a_1_22
- b_2_82 + a_2_7·a_1_2·b_1_1 + c_2_9·b_1_12 + c_2_10·a_1_22
- a_2_7·b_2_8 + c_2_9·a_1_2·b_1_1
- a_1_0·a_3_12 + a_2_7·a_1_2·b_1_1
- a_1_2·a_3_12
- a_1_0·b_3_23
- b_1_1·a_3_12 + a_1_2·b_3_23
- a_2_7·a_3_12
- b_2_8·a_3_12 + a_2_7·b_3_23
- b_1_12·b_3_23 + b_4_41·b_1_1 + b_2_8·b_3_23 + a_2_7·b_1_13 + c_2_9·a_1_2·b_1_12
- b_4_41·a_1_0
- a_1_2·b_1_1·b_3_23 + b_4_41·a_1_2 + b_2_8·a_3_12
- a_3_122
- a_3_12·b_3_23 + c_2_10·a_1_2·b_1_13
- b_2_8·b_1_1·b_3_23 + b_2_8·b_4_41 + c_2_9·b_1_1·b_3_23 + c_2_9·a_1_2·b_1_13
+ a_2_7·c_2_9·b_1_12
- b_3_232 + b_4_41·a_1_2·b_1_1 + a_2_7·b_1_1·b_3_23 + c_2_10·b_1_14 + c_4_45·a_1_22
- a_2_7·b_1_1·b_3_23 + a_2_7·b_4_41 + c_2_9·a_1_2·b_3_23 + a_2_7·c_2_10·a_1_2·b_1_1
+ a_2_7·c_2_9·a_1_2·b_1_1
- b_4_41·b_3_23 + b_4_41·a_1_2·b_1_12 + a_2_7·b_4_41·b_1_1 + c_2_10·b_1_15
+ b_2_8·c_2_10·b_1_13 + a_2_7·c_4_45·a_1_2
- b_4_41·a_3_12 + c_2_10·a_1_2·b_1_14 + a_2_7·c_2_10·b_1_13
- b_4_412 + b_4_41·a_1_2·b_1_13 + a_2_7·b_4_41·b_1_12 + c_2_10·b_1_16
+ c_2_9·c_2_10·b_1_14 + c_2_9·c_4_45·a_1_22
Data used for Benson′s test
- Benson′s completion test succeeded in degree 8.
- 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_3, a Duflot regular element of degree 1
- c_2_9, a Duflot regular element of degree 2
- c_2_10, a Duflot regular element of degree 2
- c_4_45, a Duflot regular element of degree 4
- b_1_12, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 4, 6].
- 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 4
- a_1_0 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- a_2_7 → 0, an element of degree 2
- b_2_8 → 0, an element of degree 2
- c_2_9 → c_1_32, an element of degree 2
- c_2_10 → c_1_22, an element of degree 2
- a_3_12 → 0, an element of degree 3
- b_3_23 → 0, an element of degree 3
- b_4_41 → 0, an element of degree 4
- c_4_45 → c_1_14, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 5
- a_1_0 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_1 → c_1_4, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- a_2_7 → 0, an element of degree 2
- b_2_8 → c_1_3·c_1_4, an element of degree 2
- c_2_9 → c_1_32, an element of degree 2
- c_2_10 → c_1_22, an element of degree 2
- a_3_12 → 0, an element of degree 3
- b_3_23 → c_1_2·c_1_42, an element of degree 3
- b_4_41 → c_1_2·c_1_43 + c_1_2·c_1_3·c_1_42, an element of degree 4
- c_4_45 → c_1_12·c_1_42 + c_1_14, an element of degree 4
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