S4 in S5

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) symmetric group:S4 and the group is (up to isomorphism) symmetric group:S5 (see subgroup structure of symmetric group:S5).
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Definition

The group $G$ is taken as symmetric group:S5: the symmetric group of degree five. For concreteness, we take $G$ as the symmetric group on the set ${1, 2, 3, 4, 5}$ .

We take $H = H_{5}$ as the subgroup fixing ${5}$ , so $H$ is symmetric group:S4 acting on the set ${1, 2, 3, 4}$ .

$H$ has four other conjugate subgroups, each corresponding to a different fixed point:

$H_{1}$ is the subgroup fixing ${1}$ , and is the symmetric group on the set ${2, 3, 4, 5}$ .
$H_{2}$ is the subgroup fixing ${2}$ , and is the symmetric group on the set ${1, 3, 4, 5}$ .
$H_{3}$ is the subgroup fixing ${3}$ , and is the symmetric group on the set ${1, 2, 4, 5}$ .
$H_{4}$ is the subgroup fixing ${4}$ , and is the symmetric group on the set ${1, 2, 3, 5}$ .
$H = H_{5}$ is the subgroup fixing ${5}$ , and is the symmetric group on the set ${1, 2, 3, 4}$ .

Arithmetic functions

Function	Value	Explanation
order of group	120
order of subgroup	24
index of subgroup	5
size of conjugacy class of subgroup	5	conjugacy class is made of those listed above
number of conjugacy classes in automorphism class of subgroup	1
size of automorphism class of subgroup	5

Subgroup properties

Other properties

Property	Meaning	Satisfied?	Explanation
Hall subgroup	order and index are relatively prime	Yes	${2, 3}$ -Hall subgroup. Also, order (24) and index (5) are relatively prime.
normal subgroup		No
p-complement	complement of a $p$ -Sylow subgroup	Yes	$p$ -complement for $p = 5$

GAP implementation

The group-subgroup pair can be constructed as follows:

G := SymmetricGroup(5); H := SymmetricGroup(4);