D8 ≅ C4⋊C2

D8 (presented as < a,b | a⁴=1=b², abab=1 >) has C4 (with elements {1,a,a²,a³}) as a normal subgroup. The quotient group is C2. Therefore D8 is an extension of C4 by C2. This page shows how.

The automorphism group of C4 is C2. There is a unique non-trivial automorphism from C2 (the quotient group) to C2 (the automorphism group of the normal subgroup): the non-identity member of the C2 performs the map a↔a³. This is sufficient to define the extension.

This is a sub-page of Groups of order 8, regarded as Extensions
which describes various kinds of group extensions.