Extensions for trivial outer action of V4 on Q8
This article describes all the group extensions corresponding to a particular outer action with normal subgroup quaternion group and quotient group Klein four-group.
We consider here the group extensions where the base normal subgroup is quaternion group, the quotient group is Klein four-group, and the induced outer action of the quotient group on the normal subgroup is trivial.
Description in terms of cohomology groups
We have the induced outer action which is trivial:
Composing with the natural mapping , we get a trivial map:
Thus, the extensions for the trivial outer action of on correspond to the elements of the second cohomology group for trivial group action:
The correspondence is as follows: an element of gives an extension with base and quotient . We take the central product of this extension group with , identifying the common .
See second cohomology group for trivial group action of V4 on Z2, which is isomorphic to elementary abelian group:E8.
The table below builds on the knowledge of second cohomology group for trivial group action of V4 on Z2.
|Cohomology class type||Number of cohomology classes||Corresponding group extension for on||Second part of GAP ID (order is 8)||Corresponding group extension for on (obtaining by taking a central product of the extension of on by , with identified)||Second part of GAP ID (order is 32)||Is the extension a semidirect product of by ?||Is the base characteristic in the semidirect product?||Nilpotency class of whole group||Derived length of whole group||Minimum size of generating set of whole group|
|trivial||1||elementary abelian group:E8||5||direct product of Q8 and V4||47||Yes||No||2||2||4|
|symmetric and nontrivial||3||direct product of Z4 and Z2||2||direct product of SmallGroup(16,13) and Z2||48||Yes||No||2||2||4|
|non-symmetric||3||dihedral group:D8||3||central product of D8 and Q8||50||No||No||2||2||4|
|non-symmetric||1||quaternion group||4||inner holomorph of D8||49||Yes||No||2||2||4|