Q8 ≅ C4⋊↑C2

Q8 has C4 as a normal subgroup. The quotient group is C2. Therefore Q8 is an extension of C4 by C2. This page shows how. This is neither a semidirect product, nor a central extension, it combines the features of both.

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 two maps, one a non-homomorphic map from H×H to N, and the other a homomorphism from H to aut(N).

The map from H×H to N 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. Thus b*b = a².

The homomorphism from H to aut(N) is the obvious and only non-trivial one (each is isomorphic to C2). The automorphism it performs is a↔a³. Thus b*a =a³b.

The 4-cycles of the resulting quaternion group are 1→a→a²→a³→1; 1→b→a²→a²b→1; 1→ab→a²→a³b→1

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