Nontrivial semidirect product of Z4 and Z4

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

A presentation as a metacyclic group

The group, a semidirect product $Z_4 \rtimes Z_4$, can be defined by:

$G := \langle a,x \mid a^4 = x^4 = e, xax^{-1} = a^{-1} \rangle$.

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

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	16	groups with same order
prime-base logarithm of order	4	groups with same prime-base logarithm of order
max-length of a group	4		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	4		chief length equals prime-base logarithm of order for group of prime power order
composition length	4		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	4	groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent	2	groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class	2	groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
derived length	2	groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length
Frattini length	2	groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group	2	groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group
rank of a p-group	2	groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group
normal rank of a p-group	2	groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group
characteristic rank of a p-group	2	groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group
number of subgroups	15	groups with same order and number of subgroups | groups with same prime-base logarithm of order and number of subgroups | groups with same number of subgroups
number of conjugacy classes of subgroups	13	groups with same order and number of conjugacy classes of subgroups | groups with same prime-base logarithm of order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups
number of conjugacy classes	10	groups with same order and number of conjugacy classes | groups with same prime-base logarithm of order and number of conjugacy classes | groups with same number of conjugacy classes

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.

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 ]