Semidirect product of Z16 and Z4 of semidihedral type
This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]
This group is defined as the external semidirect product where the base (acted upon) group is cyclic group:Z16 and the acting group is cyclic group:Z4, and the generator of the acting group acts via the power map.
Equivalently, it is given by the presentation:
This finite group has order 64 and has ID 48 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(64,48);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [64,48]
or just do:
to have GAP output the group ID, that we can then compare to what we want.
Description by presentation
gap> F := FreeGroup(2); <free group on the generators [ f1, f2 ]> gap> G := F/[F.1^(16),F.2^4,F.2*F.1*F.2^(-1)*F.1^(-7)]; <fp group on the generators [ f1, f2 ]> gap> IdGroup(G); [ 64, 48 ]