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Mod-2-Cohomology of L2(8).3, a group of order 1512
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
General information on the group
- L2(8).3 is a group of order 1512.
- The group order factors as
23 · 33 · 7 . - The group is defined by Group([(1,2)(3,5)(4,6)(7,9),(2,3,4)(6,7,8)]).
- It is non-abelian.
- It has 2-Rank 3.
- The centre of a Sylow 2-subgroup has rank 3.
- Its Sylow 2-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.
Structure of the cohomology ring
This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(168,43); GF(2)).General information
- The cohomology ring is of dimension 3 and depth 3.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1)·((1 − t + t2 − t3 + t4) · (1 − t2 + t4)) ( − 1 + t)3 · (1 + t + t2) · (1 + t + t2 + t3 + t4 + t5 + t6) - The a-invariants are -∞,-∞,-∞,-3. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
- 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 |
Ring generators
The cohomology ring has 5 minimal generators of maximal degree 7:
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c_3_0 , a Duflot element of degree 3 -
c_4_0 , a Duflot element of degree 4 -
c_5_0 , a Duflot element of degree 5 -
c_6_1 , a Duflot element of degree 6 -
c_7_1 , a Duflot element of degree 7
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Ring relations
There are 2 minimal relations of maximal degree 12:
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c_5_02 + c_3_0·c_7_1 + c_4_0·c_6_1 + c_4_0·c_3_02 -
c_5_0·c_7_1 + c_6_12 + c_6_1·c_3_02 + c_4_03 + c_3_04
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Data used for the Hilbert-Poincaré test
- We proved completion in degree 12 using the Hilbert-Poincaré criterion.
- 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_3_0 , an element of degree 3c_4_0 , an element of degree 4c_7_1 , an element of degree 7
- The above filter regular HSOP forms a Duflot regular sequence.
- The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 11].
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
Restriction maps
Expressing the generators as elements of H*(SmallGroup(168,43); GF(2))
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c_3_0 →c_3_0 -
c_4_0 →c_4_0 -
c_5_0 →c_5_0 -
c_6_1 →c_6_1 -
c_7_1 →c_7_1
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 3
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c_3_0 →c_1_23 + c_1_12·c_1_2 + c_1_13 + c_1_0·c_1_1·c_1_2 + c_1_0·c_1_12 + c_1_02·c_1_2
+ c_1_03 , an element of degree 3 -
c_4_0 →c_1_24 + c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2
+ c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2 + c_1_02·c_1_12 + c_1_04 , an element of degree 4 -
c_5_0 →c_1_25 + c_1_14·c_1_2 + c_1_15 + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_14
+ c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_2 + c_1_05 , an element of degree 5 -
c_6_1 →c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_14·c_1_2
+ c_1_02·c_1_24 + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_14 + c_1_04·c_1_22
+ c_1_04·c_1_1·c_1_2 + c_1_04·c_1_12 , an element of degree 6 -
c_7_1 →c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24
+ c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2 , an element of degree 7
| About the group | Ring generators | Ring relations | Completion information | Restriction maps |
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Simon King Department of Mathematics and Computer Science Friedrich-Schiller-Universität Jena 07737 Jena GERMANY
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