Second cohomology group for trivial group action of Q8 on V4

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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 quaternion group on Klein four-group. The elements of this classify the group extensions with Klein four-group in the center and quaternion group the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
Get more specific information about quaternion group |Get more specific information about 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 the quaternion group (of order 8) and A is the Klein four-group V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2.

The group is isomorphic to elementary abelian group:E16.


Cohomology class type Number of cohomology classes Corresponding group extension Nilpotency class of group Second part of GAP ID (order is 32)
trivial 1 direct product of Q8 and V4 2 47
nontrivial  ? SmallGroup(32,2) 2 2
nontrivial  ? direct product of SmallGroup(16,4) and Z2 2 23