Extensions for trivial outer action of Z2 on SL(2,3)
This article describes all the group extensions corresponding to a particular outer action with normal subgroup special linear group:SL(2,3) and quotient group cyclic group:Z2.
We consider here the group extensions where the base normal subgroup is special linear group:SL(2,3) (order 24), the quotient group is cyclic group:Z2 (order 2), 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 48)||Is the extension a semidirect product of by ?||Is the base characteristic in the whole group?||Derived length of whole group|
|trivial||1||Klein four-group||2||direct product of SL(2,3) and Z2||32||Yes||Yes||3|
|nontrivial||1||cyclic group:Z4||1||central product of SL(2,3) and Z4||13||Yes||Yes||3|