Derived subgroup of nontrivial semidirect product of Z4 and Z4

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) cyclic group:Z2 and the group is (up to isomorphism) nontrivial semidirect product of Z4 and Z4 (see subgroup structure of nontrivial semidirect product of Z4 and Z4).
The subgroup is a normal subgroup and the quotient group is isomorphic to direct product of Z4 and Z2.
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Definition

Consider the group:

$G := \langle x,y \mid x^4 = y^4 = e, yxy^{-1} = x^3 \rangle$.

This is a group of order 16 with elements:

$\! \{ e,x,x^2,x^3,y,xy,x^2y,x^3y,y^2,xy^2,x^2y^2,x^3y^2,y^3,xy^3,x^2y^3,x^3y^3 \}$

We are interested in the subgroup:

$\! H = \{ e, x^2 \}$

This is the derived subgroup. In particular, it is a normal subgroup and the quotient group is isomorphic to direct product of Z4 and Z2.

Cosets

The subgroup is a normal subgroup, so its left cosets coincide with its right cosets. The subgroup has order 2 and index 8, so there are 8 cosets, given as:

$\! \{ e, x^2 \}, \{ x , x^3 \}, \{ y, x^2y \}, \{ xy, x^3y \}, \{ y^2, x^2y^2 \}, \{xy^2, x^3y^2 \}, \{ y^3, x^2y^3 \}, \{ xy^3, x^3y^3 \}$

Complements

The subgroup has no permutable complements. Since it is a normal subgroup, this also means it has no lattice complements.

Properties related to complementation

Property	Meaning	Satisfied?	Explanation	Comment
complemented normal subgroup	normal subgroup with permutable complement	No	see above
permutably complemented subgroup	subgroup with permutable complement	No
lattice-complemented subgroup	subgroup with lattice complement	No
retract	has a normal complement	No
direct factor	normal subgroup with normal complement	No

Arithmetic functions

Function	Value	Explanation
order of whole group	16
order of subgroup	2
index	8
size of conjugacy class	1
number of conjugacy classes in automorphism class	1

Subgroup-defining functions

Subgroup-defining function	Meaning in general	Why it takes this value
derived subgroup	subgroup generated by all commutators	The commutators are precisely the elements of this subgroup.

Subgroup properties

Invariance under automorphisms and endomorphisms

Property	Meaning	Satisfied?	Explanation
normal subgroup	invariant under inner automorphisms	Yes	derived subgroup is normal
characteristic subgroup	invariant under all automorphisms	Yes	derived subgroup is characteristic
fully invariant subgroup	invariant under all endomorphisms	Yes	derived subgroup is fully invariant
verbal subgroup	generated by set of words	Yes	derived subgroup is verbal
characteristic-isomorph-free subgroup	no other isomorphic characteristic subgroup	No	$\{ y^2, e \}$ and $\{ x^2y^2, e\}$ are isomorphic characteristic subgroups. Hence also not a isomorph-free subgroup, isomorph-containing subgroup, or normal-isomorph-free subgroup.
isomorph-characteristic subgroup	Every isomorphic subgroup is characteristic	Yes	The two other isomorphic subgroups are both characteristic, see central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4 and subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4.
isomorph-normal subgroup	Every isomorphic subgroup is normal	Yes	Follows from being isomorph-characteristic.

Centrality and related properties

Property	Meaning	Satisfied?	Explanation
central subgroup	contained in the center	Yes	The center is $\langle x^2, y^2 \rangle$.
central factor	product with centralizer is whole group	Yes	central implies central factor

GAP implementation

The group and subgroup can be constructed as follows, using the SmallGroup and DerivedSubgroup functions:

G := SmallGroup(16,4); H := DerivedSubgroup(G);

Implementing this in GAP looks as follows:

gap> G := SmallGroup(16,4); H := DerivedSubgroup(G);
<pc group of size 16 with 4 generators>
Group([ f3 ])

Below is GAP implementation to test some of the assertions in this page:[SHOW MORE]

gap> StructureDescription(G/H);
"C4 x C2"
gap> Order(G);
16
gap> Order(H);
2
gap> Index(G,H);
8
gap> IsNormal(G,H);
true
gap> IsCharacteristicSubgroup(G,H);
true
gap> IsFullinvariant(G,H);
true