D8 in S4

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) symmetric group:S4 (see subgroup structure of symmetric group:S4).
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This article is about the subgroup $H$ in the group $G$, where $G$ is symmetric group:S4, i.e., the symmetric group on the set $\{ 1,2,3,4 \}$, and $H$ is the subgroup:

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

$H$ is a 2-Sylow subgroup of $G$ and is isomorphic to dihedral group:D8. It has two other conjugate subgroups, which are given below:

$\! H_1 = \{ (), (1,3,2,4), (1,2)(3,4), (1,4,2,3), (1,3)(2,4), (1,4)(2,3), (1,2), (3,4) \}$

and:

$\! H_2 = \{ (), (1,2,4,3), (1,4)(2,3), (1,3,4,2), (1,2)(3,4), (1,3)(2,4), (1,4), (2,3) \}$

Complements

The permutable complements to $H$ (and also to each of its conjugates) are precisely the subgroups of order three, namely A3 in S4 and its conjugates. Each of these is a conjugate to each of the conjugates of $H$:

$\! \{ (), (1,2,3), (1,3,2) \}, \{ () (1,3,4), (1,4,3) \}, \{ (), (1,2,4), (1,4,2) \}, \{ (), (2,3,4), (2,4,3) \}$

In addition, any S2 in S4 is a lattice complement but not a permutable complement to two of the three conjugates of $H$, namely the ones that do not contain it.

Properties related to complementation

Property	Meaning	Satisfied?	Explanation	Comment
permutably complemented subgroup	has a permutable complement	Yes	A3 in S4
lattice-complemented subgroup	has a lattice complement	Yes	A3 in S4, or S2 in S4
retract	has a normal complement	No
complemented normal subgroup		No	not a normal subgroup

Arithmetic functions

Function	Value	Explanation	Comment
order of the group	24
order of the subgroup	8
index of the subgroup	3
size of conjugacy class	3
number of conjugacy classes in automorphism class	1

Effect of subgroup operators

In the table below, we provide values specific to $H$.

Function	Value as subgroup (descriptive)	Value as subgroup (link)	Value as group
normalizer	the subgroup itself	(current page)	dihedral group:D8
centralizer	$\{ (), (1,3)(2,4) \}$	subgroup generated by double transposition in S4	cyclic group:Z2
normal core	$\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$	normal Klein four-subgroup of symmetric group:S4	Klein four-group
normal closure	the whole group	--	symmetric group:S4
characteristic core	$\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$	normal Klein four-subgroup of symmetric group:S4	Klein four-group
characteristic closure	the whole group	--	symmetric group:S4

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 on subgroup

The subgroup embedding induces the non-inner non-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 symmetric group:S4. 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 SymmetricGroup and SylowSubgroup functions as follows:

G := SymmetricGroup(4); H := SylowSubgroup(G,2);

Note that this doesn't output the subgroup $H$ used on this page, but rather the conjugate subgroup $H_1$. However all relevant properties are invariant under conjugation, so this does not matter.