Second cohomology group for trivial group action of S3 on Z2

From Groupprops
Jump to: navigation, search
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:S3 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and symmetric group:S3 the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
Get more specific information about symmetric group:S3 |Get more specific information about cyclic group:Z2

Description of the group

This article describes the second cohomology group for trivial group action:

\! H^2(G;A)

where G is symmetric group:S3 (i.e., the symmetric group on a set of size three) and A is cyclic group:Z2.

The cohomology group is isomorphic to cyclic group:Z2.

Computation of cohomology group

The cohomology group can be computed directly from group cohomology of symmetric group:S3#Cohomology groups for trivial group action.

Elements

Cohomology class type Number of cohomology classes Corresponding group extension Second part of GAP ID (order is 12)
trivial 1 dihedral group:D12 (same as direct product of S3 and Z2) 4
nontrivial 1 dicyclic group:Dic12 1