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Mod-7-Cohomology of J2, a group of order 604800
General information on the group
- J2 is a group of order 604800.
- The group order factors as 27 · 33 · 52 · 7.
- The group is defined by Group([(1,84)(2,20)(3,48)(4,56)(5,82)(6,67)(7,55)(8,41)(9,35)(10,40)(11,78)(12,100)(13,49)(14,37)(15,94)(16,76)(17,19)(18,44)(21,34)(22,85)(23,92)(24,57)(25,75)(26,28)(27,64)(29,90)(30,97)(31,38)(32,68)(33,69)(36,53)(39,61)(42,73)(43,91)(45,86)(46,81)(47,89)(50,93)(51,96)(52,72)(54,74)(58,99)(59,95)(60,63)(62,83)(65,70)(66,88)(71,87)(77,98)(79,80),(1,80,22)(2,9,11)(3,53,87)(4,23,78)(5,51,18)(6,37,24)(8,27,60)(10,62,47)(12,65,31)(13,64,19)(14,61,52)(15,98,25)(16,73,32)(17,39,33)(20,97,58)(21,96,67)(26,93,99)(28,57,35)(29,71,55)(30,69,45)(34,86,82)(38,59,94)(40,43,91)(42,68,44)(46,85,89)(48,76,90)(49,92,77)(50,66,88)(54,95,56)(63,74,72)(70,81,75)(79,100,83)]).
- It is non-abelian.
- It has 7-Rank 1.
- The centre of a Sylow 7-subgroup has rank 1.
- Its Sylow 7-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 1.
Structure of the cohomology ring
This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(42,1); GF(7)).
General information
- The cohomology ring is of dimension 1 and depth 1.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1)·(1 − t + t2 − t3 + t4 − t5 + t6 − t7 + t8 − t9 + t10) |
| ( − 1 + t) · (1 − t + t2) · (1 + t2) · (1 + t + t2) · (1 − t2 + t4) |
- The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
- The filter degree type of any filter regular HSOP is [-1, -1].
Ring generators
The cohomology ring has 2 minimal generators of maximal degree 12:
- a_11_0, a nilpotent element of degree 11
- c_12_0, a Duflot element of degree 12
Ring relations
There is one "obvious" relation:
a_11_02
Apart from that, there are no relations.
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_12_0, an element of degree 12
- The above filter regular HSOP forms a Duflot regular sequence.
- The Raw Filter Degree Type of the filter regular HSOP is [-1, 11].
Restriction maps
- a_11_0 → a_11_0
- c_12_0 → c_12_0
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1
- a_11_0 → c_2_05·a_1_0, an element of degree 11
- c_12_0 → c_2_06, an element of degree 12
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