C8 ≅ C4↑C2

C8 has C4 as a normal subgroup. The quotient group is C2. Therefore C8 is an extension of C4 by C2. This page shows how.

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 a, which goes into the normal subgroup.

Thus b has order 8, its powers being 1,b,a,ab,a²,a²b,a³,a³b.


C4↑C2

This extension can be regarded as a toll-bean extension, as shown to the left.

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