which describes various kinds of group extensions.
D8, presented as < a,b | a4=1=b2, abab=1 >, has C22, with elements {1, a2, b, a2b}, as a normal subgroup. The quotient group is C2. Therefore D8 is an extension of C22 by C2. This page shows how. This is a semidirect product.
The extension is defined by a map from C2 to aut(C22). Aut(C22) is D6, which has C2 as a subgroup, in three ways. We choose that subgroup of the automorphism group that performs the map b↔a2b.
This is a sub-page of Groups of order 8, regarded as Extensions
which describes various kinds of group extensions.
See also my main index page for groups.
Copyright N.S.Wedd 2008