Simon King
David J. Green
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Cohomology of group number 26 of order 128
General information on the group
- The group has 2 minimal generators and exponent 8.
- 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
t3 + t2 + 1 |
| (t + 1)2 · (t − 1)4 · (t2 + 1) |
- The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 15 minimal generators of maximal degree 5:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- a_2_0, a nilpotent element of degree 2
- a_2_2, a nilpotent element of degree 2
- b_2_3, an element of degree 2
- c_2_1, a Duflot regular element of degree 2
- c_2_4, a Duflot regular element of degree 2
- a_3_5, a nilpotent element of degree 3
- a_3_6, a nilpotent element of degree 3
- a_3_7, a nilpotent element of degree 3
- b_3_8, an element of degree 3
- a_4_5, a nilpotent element of degree 4
- a_4_13, a nilpotent element of degree 4
- c_4_14, a Duflot regular element of degree 4
- a_5_23, a nilpotent element of degree 5
Ring relations
There are 65 minimal relations of maximal degree 10:
- a_1_02
- a_1_12
- a_1_0·a_1_1
- a_2_0·a_1_1
- a_2_0·a_1_0
- a_2_2·a_1_1
- a_2_2·a_1_0
- b_2_3·a_1_0
- a_2_02
- a_2_22
- a_1_1·a_3_5 + a_2_0·a_2_2
- a_1_0·a_3_5
- a_1_1·a_3_6
- a_1_0·a_3_6 + a_2_0·a_2_2
- a_1_1·a_3_7
- a_1_0·a_3_7
- a_1_1·b_3_8 + a_2_0·b_2_3
- a_1_0·b_3_8 + a_2_0·a_2_2
- a_2_2·a_3_5
- a_2_0·a_3_5
- a_2_2·a_3_6
- a_2_0·a_3_6
- b_2_3·a_3_6 + a_2_2·a_3_7
- a_2_0·a_3_7
- b_2_3·a_3_5 + a_2_2·b_3_8
- a_2_0·b_3_8 + b_2_3·c_2_1·a_1_1
- a_4_5·a_1_1
- a_4_5·a_1_0
- b_2_3·a_3_6 + a_4_13·a_1_1
- a_4_13·a_1_0
- a_3_52
- a_3_62
- a_3_5·a_3_6
- a_3_72
- a_3_6·a_3_7
- a_3_6·b_3_8 + a_3_5·a_3_7
- a_3_5·b_3_8 + a_2_2·b_2_3·c_2_1
- b_3_82 + b_2_32·c_2_1
- a_3_7·b_3_8 + b_2_3·a_4_5
- a_3_5·a_3_7 + a_2_2·a_4_5
- a_2_0·a_4_5
- a_2_2·a_4_13
- a_3_5·a_3_7 + a_2_0·a_4_13
- a_3_5·a_3_7 + a_1_1·a_5_23
- a_1_0·a_5_23
- a_4_5·a_3_7
- a_4_5·a_3_6
- a_4_5·a_3_5 + a_2_2·c_2_1·a_3_7
- a_4_5·b_3_8 + b_2_3·c_2_1·a_3_7
- a_4_13·a_3_7
- a_4_13·a_3_6
- a_4_13·a_3_5
- a_4_13·b_3_8 + b_2_3·a_5_23 + a_2_2·b_2_3·b_3_8
- a_2_2·a_5_23
- a_2_0·a_5_23 + a_2_2·c_2_1·a_3_7
- a_4_52
- a_4_5·a_4_13 + a_2_0·a_2_2·c_4_14
- a_4_132
- a_3_7·a_5_23 + a_2_0·b_2_3·a_4_13 + a_2_0·a_2_2·c_4_14
- a_3_6·a_5_23
- a_3_5·a_5_23
- b_3_8·a_5_23 + b_2_3·c_2_1·a_4_13 + a_2_2·b_2_32·c_2_1
- a_4_5·a_5_23 + a_2_2·b_2_3·c_2_1·a_3_7
- a_4_13·a_5_23
- a_5_232
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_1, a Duflot regular element of degree 2
- c_2_4, a Duflot regular element of degree 2
- c_4_14, a Duflot regular element of degree 4
- b_2_3, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 4, 6].
- 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_1 → 0, an element of degree 1
- a_2_0 → 0, an element of degree 2
- a_2_2 → 0, an element of degree 2
- b_2_3 → 0, an element of degree 2
- c_2_1 → c_1_02, an element of degree 2
- c_2_4 → c_1_22 + c_1_12, an element of degree 2
- a_3_5 → 0, an element of degree 3
- a_3_6 → 0, an element of degree 3
- a_3_7 → 0, an element of degree 3
- b_3_8 → 0, an element of degree 3
- a_4_5 → 0, an element of degree 4
- a_4_13 → 0, an element of degree 4
- c_4_14 → c_1_24, an element of degree 4
- a_5_23 → 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_1 → 0, an element of degree 1
- a_2_0 → 0, an element of degree 2
- a_2_2 → 0, an element of degree 2
- b_2_3 → c_1_32, an element of degree 2
- c_2_1 → c_1_02, an element of degree 2
- c_2_4 → c_1_22 + c_1_12, an element of degree 2
- a_3_5 → 0, an element of degree 3
- a_3_6 → 0, an element of degree 3
- a_3_7 → 0, an element of degree 3
- b_3_8 → c_1_0·c_1_32, an element of degree 3
- a_4_5 → 0, an element of degree 4
- a_4_13 → 0, an element of degree 4
- c_4_14 → c_1_24 + c_1_12·c_1_32, an element of degree 4
- a_5_23 → 0, an element of degree 5
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