Q8 in SD16

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) quaternion group and the group is (up to isomorphism) semidihedral group:SD16 (see subgroup structure of semidihedral group:SD16).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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Definition

Here, $G$ is the semidihedral group:SD16, the semidihedral group of order sixteen (and hence, degree eight). We use here the presentation:

$G := \langle a,x \mid a^8 = x^2 = e, xax = a^3 \rangle$

$G$ has 16 elements:

$\! \{ e,a,a^2,a^3,a^4,a^5,a^6,a^7,x,ax,a^2x,a^3x,a^4x,a^5x,a^6x,a^7x \}$

The subgroup $H$ of interest is the subgroup $\langle a^2,ax \rangle$. It is a quaternion group of order 8 and is given by:

$\! \{ e,a^2,a^4,a^6,ax,a^3x,a^5x,a^7x \}$

Arithmetic functions

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

Subgroup properties

Invariance under automorphisms and endomorphisms

Property	Meaning	Satisfied?	Explanation
normal subgroup	invariant under inner automorphisms	Yes	index two implies normal
characteristic subgroup	invariant under all automorphisms	Yes	On account of being an isomorph-free subgroup.
fully invariant subgroup	invariant under all endomorphisms	No	The endomorphism $a \mapsto e, x \mapsto x$ (i.e., the retraction to $\langle x \rangle$) does not preserve this subgroup.
isomorph-free subgroup	no other isomorphic subgroup	Yes

GAP implementation

The group and subgroup pair can be constructed using GAP as follows:

gap> G := SmallGroup(16,8); H := Filtered(NormalSubgroups(G), x -> IdGroup(x) = [8,4])[1];

Here is the GAP display:

gap> G := SmallGroup(16,8); H := Filtered(NormalSubgroups(G), x -> IdGroup(x) = [8,4])[1];
<pc group of size 16 with 4 generators>
Group([ f1, f3, f4 ])

Here is GAP code to verify various assertions made on this page:

gap> Order(G);
16
gap> Order(H);
8
gap> Index(G,H);
2
gap> StructureDescription(H);
"Q8"
gap> StructureDescription(G/H);
"C2"
gap> IsNormal(G,H);
true
gap> IsCharacteristicSubgroup(G,H);
true
gap> IsFullinvariant(G,H);
false