Nontrivial semidirect product of Z4 and Z4

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Definition

A presentation as a metacyclic group

The group can be defined by:

$G:=⟨x,y\mid x^4=y^4=e,yx{y}^{-1}=x^3⟩$.

An alternative presentation

This group can be defined by the following presentation:

$G:=⟨a,b,c\mid a^2=b^4=e,ab=ba,ac=ca,b^2=c^2,cb{c}^{-1}=ab⟩$

The subgroup $⟨a,b⟩$ is isomorphic to the direct product of Z4 and Z2.

The two presentations are related by $x=bc,y=c$.

Arithmetic functions

Want to compare with other groups of the same order? Check out groups of order 16#Arithmetic functions

Function	Value	Explanation
order	16
exponent	4
Frattini length	2
Fitting length	1
derived length	2
nilpotency class	2
minimum size of generating set	2
rank as p-group	2
normal rank as p-group	2
characteristic rank as p-group	2
number of subgroups	15
number of conjugacy classes of subgroups	13
number of conjugacy classes	10

Group properties

Want to compare with other groups of the same order? Check out groups of order 16#Group properties.

Property	Satisfied	Explanation
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

Subgroups

Further information: subgroup structure of SmallGroup(16,4)

1. The trivial subgroup. Isomorphic to trivial group. (1)
2. The two-element subgroup $⟨a⟩=⟨x^2⟩$. Isomorphic to cyclic group:Z2. This can be defined as the commutator subgroup. (1)
3. The two-element subgroup $⟨b^2⟩=⟨y^2⟩$. Isomorphic to cyclic group:Z2. This can be defined as the subgroup of order two whose non-identity element is the unique element of order two that is a square but is not a commutator. (1)
4. The two-element subgroup $⟨a{b}^2⟩=⟨x^2{y}^2⟩$. Isomorphic to cyclic group:Z2. This can be defined as the unique subgroup of order two whose non-identity element is a product of squares but is not a square. (1)
5. The four-element subgroup $⟨a,b^2⟩=⟨x^2,y^2⟩$. Isomorphic to Klein four-group. (1)
6. The subgroups $⟨b⟩$, $⟨ab⟩$, $⟨c⟩$, and $⟨ac⟩$. Isomorphic to cyclic group:Z4. All of these contain the element $b^2$. None is normal. They come in two conjugacy classes: $⟨b⟩$ with $⟨ab⟩$, and $⟨c⟩$ with $⟨ac⟩$.(4)
7. The subgroups $⟨bc⟩$ and $⟨abc⟩$. Isomorphic to cyclic group:Z4. Both are normal, and are related by an outer automorphism. (2)
8. The subgroups $⟨a,b⟩$ and $⟨a,c⟩$. Isomorphic to direct product of Z4 and Z2. Both are normal and are related by an outer automorphism. (2)
9. The subgroup $⟨b^2,bc⟩$. Isomorphic to direct product of Z4 and Z2. A characteristic subgroup. (1)
10. The whole group.

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 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 ]