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.


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