Nontrivial semidirect product of Z4 and Z4
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Contents
Definition
A presentation as a metacyclic group
The group, a semidirect product , can be defined by:
.
Arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions
Group properties
Want to compare and contrast group properties with other groups of the same order? Check out groups of order 16#Group properties
Property | Satisfied? | Explanation | Comment |
---|---|---|---|
Abelian group | No | ||
Group of prime power order | Yes | ||
Nilpotent group | Yes | ||
Metabelian group | Yes | ||
Metacyclic group | Yes | ||
Supersolvable group | Yes | ||
Group of nilpotency class two | Yes | ||
T-group | No | ||
Directly indecomposable group | Yes | ||
Splitting-simple group | No | ||
UL-equivalent group | No | See also nilpotent not implies UL-equivalent |
Subgroups
Further information: subgroup structure of nontrivial semidirect product of Z4 and Z4
In case a single equivalence class of subgroups under automorphisms comprises multiple conjugacy classes of subgroups, outer curly braces are used to bucket the conjugacy classes.
GAP implementation
Group ID
This finite group has order 16 and has ID 4 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(16,4)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(16,4);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [16,4]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.
Other descriptions
gap> G := F/[F.1^2, F.2^4, F.1*F.2*F.1^(-1)*F.2^(-1),F.3^4,F.3*F.1*F.3^(-1)*F.1^(-1),F.3*F.2*F.3^(-1)*F.2^(-1)*F.1^(-1), F.3^2 * F.2^2]; <fp group on the generators [ f1, f2, f3 ]> gap> IdGroup(G); [ 16, 4 ]