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.
Extensions
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 ![]() ![]() |
Second part of GAP ID (order is 8) | Corresponding group extension for ![]() ![]() ![]() ![]() ![]() ![]() |
Second part of GAP ID (order is 32) | Is the extension a semidirect product of ![]() ![]() |
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 |
Total (--) | 8 | -- | -- | -- | -- | -- | -- | -- | -- | -- |