C4×C2 ≅ C4↑C2

C4×C2 has C4 as a normal subgroup. The quotient group is C2. Therefore C4×C2 is an extension of C4 by C2. This page shows how (another simpler way is mentioned in the page on direct products).

Call the elements of the normal subgroup N {1,a,a²,a³}. Call the elements of the quotient group H {1,b}. The extension is defined by a map from H×H to N. This map is specified by colouring the Cayley table of H like this:

*	1	b
1	1	b
b	b	1

The pink cell of this table generates the element a², which goes into the normal subgroup.

The resulting group is the direct product of {1,a,a²,a³} and {1,ab}.

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