D8 in A6

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) dihedral group:D8 and the group is (up to isomorphism) alternating group:A6 (see subgroup structure of alternating group:A6).
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This article is about the subgroup $H$ in the group $G$, where $G$ is alternating group:A6, i.e., the alternating group on the set $\{ 1,2,3,4,5,6 \}$, and $H$ is the subgroup:

$\! H = \{ (), (1,2,3,4)(5,6), (1,3)(2,4), (1,4,3,2)(5,6), (1,2)(3,4), (1,4)(2,3), (1,3)(5,6), (2,4)(5,6) \}$

$H$ is a 2-Sylow subgroup of $G$ and is isomorphic to dihedral group:D8. There is a total of 45 conjugate subgroups to $H$ in $G$ (including $H$ itself).

Arithmetic functions

Function	Value	Explanation	Comment
order of the group	360
order of the subgroup	8
index of the subgroup	45
size of conjugacy class	45
number of conjugacy classes in automorphism class	1

Subgroup properties

Resemblance-based properties

Property	Meaning	Satisfied?	Explanation	Comment
order-conjugate subgroup	conjugate to any subgroup of the same order	Yes	Sylow implies order-conjugate
order-dominating subgroup	any subgroup of the whole group whose order divides the order of $H$ is contained in a conjugate of $H$	Yes	Sylow implies order-dominating
order-dominated subgroup	any subgroup of the whole group whose order is a multiple of the order of $H$ contains a conjugate of $H$	Yes	Sylow implies order-dominated
order-isomorphic subgroup	isomorphic to any subgroup of the group of the same order	Yes	(via order-conjugate)
isomorph-automorphic subgroup		Yes	(via order-conjugate)
automorph-conjugate subgroup		Yes	(via order-conjugate)
order-automorphic subgroup		Yes	(via order-conjugate)
isomorph-conjugate subgroup		Yes	(via order-conjugate)

Normality-related properties

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	equals all its conjugate subgroups	No	(see other conjugate subgroups)
subnormal subgroup		No
self-normalizing subgroup	equals its normalizer in the whole group	Yes
abnormal subgroup		Yes
weakly abnormal subgroup		Yes
contranormal subgroup		Yes
maximal subgroup		Yes
pronormal subgroup		Yes	Sylow implies pronormal
weakly pronormal subgroup		Yes	(via pronormal)

Fusion system

The subgroup embedding induces the simple fusion system for dihedral group:D8.

GAP implementation

Construction of subgroup given group as a black box

Suppose we are already given a group $G$ that we know to be isomorphic to alternating group:A6. Then, the subgroup $H$ can be constructed using SylowSubgroup as follows:

H := SylowSubgroup(G,2);

Construction of group-subgroup pair

The group and subgroup pair can be defined using GAP's AlternatingGroup and SylowSubgroup functions as follows:

G := AlternatingGroup(6); H := SylowSubgroup(G,2);