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Cohomology of group number 1682 of order 128
| About the group | Ring generators | Ring relations | Completion information | Restriction maps | Back to groups 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) · (t4 + 1) - The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.
| About the group | Ring generators | Ring relations | Completion information | Restriction maps | Back to groups of order 128 |
Ring generators
The cohomology ring has 7 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 -
c_2_8 , a Duflot regular element of degree 2 -
b_5_26 , an element of degree 5 -
c_8_45 , a Duflot regular element of degree 8
| About the group | Ring generators | Ring relations | Completion information | Restriction maps | Back to groups of order 128 |
Ring relations
There are 6 minimal relations of maximal degree 10:
-
a_1_0·a_1_1 -
a_1_22 + a_1_02 -
a_1_2·b_1_32 + a_1_0·b_1_32 + a_1_1·a_1_2·b_1_3 + a_1_0·a_1_2·b_1_3 + a_1_02·b_1_3
+ a_1_13 + a_1_03 -
a_1_13·b_1_32 + a_1_03·b_1_32 + a_1_14·b_1_3 + a_1_04·b_1_3 + a_1_15 + a_1_05 -
a_1_2·b_5_26 + a_1_0·b_5_26 + a_1_06 + c_2_8·a_1_12·a_1_2·b_1_3
+ c_2_8·a_1_02·a_1_2·b_1_3 + c_2_8·a_1_03·b_1_3 + c_2_8·a_1_14 -
b_5_262 + c_2_8·b_1_38 + c_2_82·a_1_12·b_1_34 + c_2_82·a_1_05·b_1_3
+ c_2_82·a_1_06
| About the group | Ring generators | Ring relations | Completion information | Restriction maps | Back to groups of order 128 |
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_8 , a Duflot regular element of degree 2c_8_45 , a Duflot regular element of degree 8b_1_32 , an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 9].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps | Back to groups of order 128 |
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 2
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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 -
c_2_8 →c_1_02 , an element of degree 2 -
b_5_26 →0 , an element of degree 5 -
c_8_45 →c_1_18 , 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 -
c_2_8 →c_1_02 , an element of degree 2 -
b_5_26 →c_1_0·c_1_24 , an element of degree 5 -
c_8_45 →c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_27 + c_1_02·c_1_26 + c_1_04·c_1_24
+ c_1_06·c_1_22 , an element of degree 8
| About the group | Ring generators | Ring relations | Completion information | Restriction maps | Back to groups of order 128 |
| Simon A. King | David J. Green | ||||||||||||||||
| Fakultät für Mathematik und Informatik | Fakultät für Mathematik und Informatik | ||||||||||||||||
| Friedrich-Schiller-Universität Jena | Friedrich-Schiller-Universität Jena | ||||||||||||||||
| Ernst-Abbe-Platz 2 | Ernst-Abbe-Platz 2 | ||||||||||||||||
| D-07743 Jena | D-07743 Jena | ||||||||||||||||
| Germany | Germany | ||||||||||||||||
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