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Cohomology of group number 11 of order 64
| About the group | Ring generators | Ring relations | Completion information | Restriction maps | Back to groups of order 64 |
General information on the group
- The group has 2 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 2.
- Its center has rank 2.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 2 and depth 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
(t2 − t + 1) · (t2 + t + 1) (t − 1)2 · (t2 + 1)2 - The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Benson test.
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Ring generators
The cohomology ring has 10 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_1 , a nilpotent element of degree 2 -
a_3_0 , a nilpotent element of degree 3 -
a_3_1 , a nilpotent element of degree 3 -
a_4_0 , a nilpotent element of degree 4 -
a_4_1 , a nilpotent element of degree 4 -
c_4_2 , a Duflot regular element of degree 4 -
c_4_3 , a Duflot regular element of degree 4 -
a_5_5 , a nilpotent element of degree 5
| About the group | Ring generators | Ring relations | Completion information | Restriction maps | Back to groups of order 64 |
Ring relations
There are 35 minimal relations of maximal degree 10:
-
a_1_02 -
a_1_0·a_1_1 -
a_1_13 -
a_2_1·a_1_1 -
a_2_1·a_1_0 -
a_2_12 -
a_1_1·a_3_0 -
a_1_0·a_3_0 -
a_1_1·a_3_1 -
a_1_0·a_3_1 -
a_2_1·a_3_0 -
a_2_1·a_3_1 -
a_4_0·a_1_1 -
a_4_0·a_1_0 -
a_4_1·a_1_0 -
a_3_02 -
a_3_12 -
a_3_0·a_3_1 + a_2_1·a_4_0 -
a_3_0·a_3_1 + a_2_1·a_4_1 -
a_3_0·a_3_1 + a_4_1·a_1_12 -
a_1_1·a_5_5 + c_4_3·a_1_12 + c_4_2·a_1_12 -
a_3_0·a_3_1 + a_1_0·a_5_5 -
a_4_0·a_3_0 -
a_4_0·a_3_1 -
a_4_1·a_3_0 -
a_4_1·a_3_1 -
a_2_1·a_5_5 -
a_4_02 -
a_4_0·a_4_1 -
a_4_12 -
a_3_0·a_5_5 -
a_3_1·a_5_5 -
a_4_0·a_5_5 -
a_4_1·a_5_5 + a_4_1·c_4_3·a_1_1 + a_4_1·c_4_2·a_1_1 -
a_5_52 + c_4_32·a_1_12 + c_4_22·a_1_12
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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_4_2 , a Duflot regular element of degree 4c_4_3 , a Duflot regular element of degree 4
- The Raw Filter Degree Type of that HSOP is [-1, -1, 6].
- The filter degree type of any filter regular HSOP is [-1, -2, -2].
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Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 2
-
a_1_0 →0 , an element of degree 1 -
a_1_1 →0 , an element of degree 1 -
a_2_1 →0 , an element of degree 2 -
a_3_0 →0 , an element of degree 3 -
a_3_1 →0 , an element of degree 3 -
a_4_0 →0 , an element of degree 4 -
a_4_1 →0 , an element of degree 4 -
c_4_2 →c_1_04 , an element of degree 4 -
c_4_3 →c_1_14 + c_1_04 , an element of degree 4 -
a_5_5 →0 , an element of degree 5
| About the group | Ring generators | Ring relations | Completion information | Restriction maps | Back to groups of order 64 |
| 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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