Extensions for trivial outer action of Z2 on D8
This article describes all the group extensions corresponding to a particular outer action with normal subgroup dihedral group:D8 and quotient group cyclic group:Z2.
We consider here the group extensions where the base normal subgroup is dihedral group:D8, the quotient group is cyclic group:Z2, 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 Z2 on Z2, which is isomorphic to cyclic group:Z2.
|Cohomology class type||Number of cohomology classes||Corresponding group extension for on||Second part of GAP ID (order is 4)||Corresponding group extension for on (obtained by taking the central product with of the extension for on )||Second part of GAP ID (order is 16)||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||Klein four-group||2||direct product of D8 and Z2||11||Yes||No||2||2||3|
|nontrivial||1||cyclic group:Z4||1||central product of D8 and Z4||13||Yes||No||2||2||3|