Second cohomology group for trivial group action of V4 on direct product of Z4 and Z2

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 Klein four-group on direct product of Z4 and Z2. The elements of this classify the group extensions with direct product of Z4 and Z2 in the center and Klein four-group the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
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Description of the group

We consider here the second cohomology group for trivial group action of Klein four-group on direct product of Z4 and Z2, i.e.,

$\! H^2(G,A)$

where $G \cong V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$ and $A \cong \mathbb{Z}_4 \times \mathbb{Z}_2$.

The group is isomorphic to elementary abelian group:E64.

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 32)
trivial	1	direct product of E8 and Z4	45
one of the symmetric nontrivial ones	3	direct product of Z4 and Z4 and Z2	21
one of the symmetric nontrivial ones	6	direct product of Z8 and V4	36
one of the symmetric nontrivial ones	6	direct product of Z8 and Z4	3
non-symmetric	3	direct product of SmallGroup(16,13) and Z2	48
non-symmetric	9	direct product of M16 and Z2	37
non-symmetric	18	SmallGroup(32,24)	24
non-symmetric	18	semidirect product of Z8 and Z4 of M-type	4

Subgroups of interest

Subgroup	Value	Corresponding group extensions for subgroup	GAP IDs second part	Group extension groupings for each coset	GAP IDs second part
IIP subgroup of second cohomology group for trivial group action	Klein four-group	direct product of E8 and Z4, direct product of SmallGroup(16,13) and Z2 (3 times)	45, 48	(direct product of E8 and Z4, direct product of SmallGroup(16,13) and Z2), (direct product of Z8 and V4, direct product of M16 and Z2), (direct product of Z4 and Z4 and Z2, SmallGroup(32,24)), (direct product of Z8 and Z4, semidirect product of Z8 and Z4 of M-type)	(45,48), (36,37), (21,24), (3,4)
cyclicity-preserving subgroup of second cohomology group for trivial group action	Klein four-group	direct product of E8 and Z4, direct product of SmallGroup(16,13) and Z2 (3 times)	45, 48	(direct product of E8 and Z4, direct product of SmallGroup(16,13) and Z2), (direct product of Z8 and V4, direct product of M16 and Z2), (direct product of Z4 and Z4 and Z2, SmallGroup(32,24)), (direct product of Z8 and Z4, semidirect product of Z8 and Z4 of M-type)	(45,48), (36,37), (21,24), (3,4)
subgroup generated by images of symmetric 2-cocycles (corresponds to abelian group extensions)	elementary abelian group:E16	direct product of E8 and Z4, direct product of Z4 and Z4 and Z2, direct product of Z8 and V4, direct product of Z8 and Z4	45, 21, 36, 3	(direct product of E8 and Z4, direct product of Z4 and Z4 and Z2, direct product of Z8 and V4, direct product of Z8 and Z4), (direct product of SmallGroup(16,13) and Z2, SmallGroup(32,24), direct product of M16 and Z2, semidirect product of Z8 and Z4 of M-type)	(45,21,36,3), (48,37,24,4)

Generalized Baer Lie rings

The examples here illustrate the cocycle halving generalization of Baer correspondence. See also second cohomology group for trivial group action is internal direct sum of symmetric and cyclicity-preserving 2-cocycle subgroups if acting group is elementary abelian 2-group and every element of order two in the base group is a square

For this particular choice of $G$ and $A$, the symmetric cohomology classes (corresponding to abelian group extensions) and the cyclicity-preserving subgroup generate the whole group, i.e., we have:

$H^2(G,A) = H^2_{sym}(G,A) + H^2_{CP}(G,A)$

as an internal direct sum. In particular, every extension has a generalized Baer cyclicity-preserving Lie ring. A pictorial description of this would be as follows. Here, each column is a coset of $H^2_{CP}(G,A)$ and each row is a coset of $H^2_{sym}(G,A)$. The top left entry is the identity element, hence the top row corresponds to abelian group extensions and the left column corresponds to cyclicity-preserving 2-cocycles.

To avoid unnecessary duplication, we have compressed the table as follows: for each repeated column we have simply indicated the number of repeats in parentheses. The full table has 16 columns.

direct product of E8 and Z4 (1 time)	direct product of Z4 and Z4 and Z2 (3 times)	direct product of Z8 and V4 (6 times)	direct product of Z8 and Z4 (6 times)
direct product of SmallGroup(16,13) and Z2	SmallGroup(32,24)	direct product of M16 and Z2	semidirect product of Z8 and Z4 of M-type
direct product of SmallGroup(16,13) and Z2	SmallGroup(32,24)	direct product of M16 and Z2	semidirect product of Z8 and Z4 of M-type
direct product of SmallGroup(16,13) and Z2	SmallGroup(32,24)	direct product of M16 and Z2	semidirect product of Z8 and Z4 of M-type