Second cohomology group for trivial group action of S4 on Z2

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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 symmetric group:S4 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and symmetric group:S4 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 Klein four-group.
Get more specific information about symmetric group:S4 |Get more specific information about cyclic group:Z2|View other constructions whose value is Klein four-group

Description of the group

This article describes the second cohomology group for trivial group action

\! H^2(G;A)

where G is symmetric group:S4 (the symmetric group on a set of size four) and A is cyclic group:Z2. G itself has order 24 and A has order 2.

The cohomology group H^2(G;A) is isomorphic to the Klein four-group.

Computation of the group

See group cohomology of symmetric group:S4#Cohomology groups for trivial group action. It is clear that we have:

H^2(G;A) \cong A/2A \oplus \operatorname{Ann}_A(2)

In the case that A is cyclic of order two, both the groups A/2A and \operatorname{Ann}_A(2) are cyclic of order two, so their direct sum is the Klein four-group.


Note that in all cases, the base of the group extension is characteristic in the whole group because it is precisely the center of the whole group. This is because the quotient group is a centerless group and hence cannot partake of any of the center.

Cohomology class type Number of cohomology classes Corresponding group extension Second part of GAP ID (order is 48) Base characteristic in whole group? Derived length
trivial 1 direct product of S4 and Z2 48 Yes 3
nontrivial (but symmetric on commuting pairs? may be) 1 special linear group:SL(2,Z4) 30 Yes 3
nontrivial 1 binary octahedral group 28 Yes 4
nontrivial 1 general linear group:GL(2,3) 29 Yes 4
Total (--) 4 -- -- --