C4 × C2 ≅ C2↑C22

C4×C2 (presented as < a,b | a4=1=b2, ab=ba >) has C2 (with elements {1,a2}) as a normal subgroup. The quotient group is C22. Therefore C4×C2 is an extension of C2 by C22. This page shows how.

The elements of the normal subgroup N are {1,-1}. Call the elements of the quotient group H {1,p,q,r} so that pq=r, pr=q, qr=p. 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:

*1pqr
11pqr
pp1rq
qqr1p
rrqp1

The pink cells of this table generate the element -1, which goes into the normal subgroup.

The correspondence between the combination {1,-1},{1,p,q,r} and the more familiar {1,a,a2,a3, b,ab,a2b,a3b} is

1,11
-1,1a2
1,pa
-1,pa3
1,qab
-1,qa3b
1,rb
-1,ra2b


C2 ↑ C22
C2 ↑ C22
C2 ↑ C22
This extension can be regarded as a toll-bean extension, in all of the ways shown to the left.



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

See also my main index page for groups.

Copyright N.S.Wedd 2008