# Nontrivial semidirect product of Z4 and Z4

(Redirected from SmallGroup(16,4))
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## Definition

### A presentation as a metacyclic group

The group can be defined by: $G := \langle x,y \mid x^4 = y^4 = e, yxy^{-1} = x^3 \rangle$.

## 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

## 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.

Automorphism class of subgroups List of subgroups Isomorphism class Order of subgroups Index of subgroups Number of conjugacy classes (=1 iff automorph-conjugate subgroup) Size of each conjugacy class (=1 iff normal subgroup) Total number of subgroups (=1 iff characteristic subgroup) Isomorphism class of quotient (if exists) Subnormal depth Nilpotency class
trivial subgroup $\{ e \}$ trivial group 1 16 1 1 1 nontrivial semidirect product of Z4 and Z4 1 0
derived subgroup of nontrivial semidirect product of Z4 and Z4 $\{ e, x^2 \}$ cyclic group:Z2 2 8 1 1 1 direct product of Z4 and Z2 1 1
subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4 $\{ e, y^2 \}$ cyclic group:Z2 2 8 1 1 1 dihedral group:D8 1 1
central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4 $\{ e, x^2y^2 \}$ cyclic group:Z2 2 8 1 1 1 quaternion group 1 1
center of nontrivial semidirect product of Z4 and Z4 $\{ e, x^2, y^2, x^2y^2 \}$ Klein four-group 4 4 1 1 1 Klein four-group 1 1
a bunch of cyclic subgroups of order four $\{ \langle y \rangle, \langle x^2 y \rangle \}$ $\{ \langle xy \rangle, \langle x^3y \rangle \}$
cyclic group:Z4 4 4 2 2 4 -- 2 1
another bunch of cyclic subgroups of order four $\langle x \rangle, \langle xy^2 \rangle$ cyclic group:Z4 4 4 2 1 2 cyclic group:Z4 1 1
abelian maximal subgroups that are not characteristic $\langle x^2, y \rangle$, $\langle x^2, xy \rangle$ direct product of Z4 and Z2 8 2 2 1 2 cyclic group:Z2 1 1
abelian maximal subgroup that is characteristic $\langle x,y^2 \rangle$ direct product of Z4 and Z2 8 2 1 1 1 cyclic group:Z2 1 1
whole group $\langle x,y \rangle$ nontrivial semidirect product of Z4 and Z4 16 1 1 1 1 trivial group 1 1
Total -- -- -- -- 13 -- 15 -- -- --

## 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 $G$:

gap> G := SmallGroup(16,4);

Conversely, to check whether a given group $G$ 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 ]