Second cohomology group for trivial group action of S5 on 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 symmetric group:S5 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and symmetric group:S5 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:S5 |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:S5 (the symmetric group of degree five) and $A$ is cyclic group:Z2. Note that $G$ has order 120 and $A$ has order 2, so all the corresponding extensions have order $120\times 2=240$.

The cohomology group itself is isomorphic to the Klein four-group.

Computation of the group

The group can be computed using group cohomology of symmetric group:S5#Cohomology groups for trivial group action. As per this, we have:

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

For $A$ cyclic of order two, both $A/2A$ and $\operatorname {Ann} _{A}(2)$ are cyclic of order two.

Elements

Note that in all cases, the base of the group extension (isomorphic to cyclic group:Z2) equals the center of the group extension and hence is a characteristic subgroup.

Cohomology class type	Number of cohomology classes	Corresponding group extension	Second part of GAP ID (order is 240)	Base characteristic in whole group?
trivial	1	direct product of S5 and Z2	189	Yes
nontrivial	1	?	89	Yes
nontrivial	1	?	90	Yes
nontrivial	1	?	91	Yes