Second cohomology group for trivial group action of direct product of Z4 and Z4 on Z4

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This article gives information about the second cohomology group for trivial group action (i.e., the second cohomology group with trivial action) of the group direct product of Z4 and Z4 on cyclic group:Z4. The elements of this classify the group extensions with cyclic group:Z4 in the center and direct product of Z4 and Z4 the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
The value of this cohomology group is direct product of Z4 and Z4 and Z4.
Get more specific information about direct product of Z4 and Z4 |Get more specific information about cyclic group:Z4|View other constructions whose value is direct product of Z4 and Z4 and Z4

Description of the group

This article is about the second cohomology group for trivial group action where the acting group is direct product of Z4 and Z4 and the group acted upon is cyclic group:Z4. In other words, it is about the cohomology group:

H^2(G,A)

where G \cong \mathbb{Z}_4 \times \mathbb{Z}_4 and A \cong \mathbb{Z}_4.

The cohomology group is isomorphic to direct product of Z4 and Z4 and Z4. In other words, it is a homocyclic group of order 64 and exponent 4.

Elements

We list here the elements, grouped by similarity under the action of the automorphism groups on both sides.

Cohomology class type Number of cohomology classes Corresponding group extension Second part of GAP ID (order is 64)
trivial 1 direct product of Z4 and Z4 and Z4 55
symmetric, nontrivial, order two 3 direct product of Z8 and Z4 and Z2 83
symmetric, nontrivial, order four 12 direct product of Z16 and Z4 26
one of the non-symmetric ones, order four 2 unitriangular matrix group:UT(3,Z4) 18
one of the non-symmetric ones, order two 1 SmallGroup(64,57) 57
one of the non-symmetric ones 6 SmallGroup(64,19) 19
one of the non-symmetric ones 3 direct product of SmallGroup(32,4) and Z2 84
one of the non-symmetric ones 24 semidirect product of Z16 and Z4 via fifth power map 28
one of the non-symmetric ones 12 semidirect product of Z16 and Z4 of M-type 27

Subgroups of interest

Subgroup Value Corresponding group extensions for subgroup GAP IDs second part Group extension groupings for each coset GAP IDs second part
cyclicity-preserving subgroup of second cohomology group for trivial group action cyclic group:Z2 direct product of Z4 and Z4 and Z4, SmallGroup(64,57) 55, 57 (direct product of Z4 and Z4 and Z4, SmallGroup(64,57)), (direct product of Z8 and Z4 and Z2, direct product of SmallGroup(32,4) and Z2) (3 times),
(direct product of Z16 and Z4, semidirect product of Z16 and Z4 of M-type) (12 times)
(55, 57), (83, 84), (26, 27)
subgroup generated by images of symmetric 2-cocycles (corresponds to abelian group extensions) direct product of Z4 and Z4 direct product of Z4 and Z4 and Z4, direct product of Z8 and Z4 and Z2 (3 times), direct product of Z16 and Z4 (12 times) 55, 83, 26 (direct product of Z4 and Z4 and Z4, direct product of Z8 and Z4 and Z2 (3 times), direct product of Z16 and Z4 (12 times))
(unitriangular matrix group:UT(3,Z4), SmallGroup(64,19) (3 times), semidirect product of Z16 and Z4 via fifth power map (12 times)) (2 times),
(SmallGroup(64,57), direct product of SmallGroup(32,4) and Z2 (3 times), semidirect product of Z16 and Z4 of M-type (12 times))
(55, 83, 26), (18, 19, 28), (57, 84, 27)

Generalized Baer Lie rings

For this particular choice of G and A, the symmetric cohomology classes (which correspond to abelian group extensions) and the cyclicity-preserving subgroup do not generate everything, but rather, they generate a subgroup of order 32 and index two, given as two cosets of the symmetric cohomology classes, giving 16 \times 2 = 32 of the 64 possible extensions. The table below shows these generalized Baer Lie rings, with each row representing a coset of the subgroup of symmetric cohomology classes and each column a coset of the subgroup of cohomology classes with a cyclicity-preserving representative. The first row and first column give the subgroup of symmetric coohmology classes and the cyclicity-preserving subgroup respectively. Note that to avoid too many columns, repeated columns are simply indicated by specifying how many columns there are of the type:

This is the smallest case where the subgroup represented by cyclicity-preserving cocycles actually has the property that there are nonzero additive cyclicity-preserving cocycles, i.e, they are linear. Thus, the generalized Baer Lie rings that we obtain in fact arise via the linear halving generalization of Baer correspondence.
direct product of Z4 and Z4 and Z4 direct product of Z8 and Z4 and Z2 (3 times) direct product of Z16 and Z4 (12 times)
SmallGroup(64,57) direct product of SmallGroup(32,2) and Z2 semidirect product of Z16 and Z4 of M-type

Here now is the full table with all cohomology classes, including the ones that are not cyclicity-preserving:

direct product of Z4 and Z4 and Z4 direct product of Z8 and Z4 and Z2 (3 times) direct product of Z16 and Z4 (12 times)
unitriangular matrix group:UT(3,Z4) SmallGroup(64,19) semidirect product of Z16 and Z4 via fifth power map
unitriangular matrix group:UT(3,Z4) SmallGroup(64,19) semidirect product of Z16 and Z4 via fifth power map
SmallGroup(64,57) direct product of SmallGroup(32,2) and Z2 semidirect product of Z16 and Z4 of M-type