C8 ≅ C2↑C4

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

Call the elements of the normal subgroup N {1,a}. Call the elements of the quotient group H {1,b,b²,b³}. 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	b²	b³
1	1	b	b²	b³
b	b	b²	b³	1
b²	b²	b³	1	b
b³	b³	1	b	b²

The pink cells of this table generate the element a, which goes into the normal subgroup.

Thus b has order 8, its powers being 1,b,b²,b³,a,ab,ab²,ab³.


C2↑C4

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.