S3 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:S3 and the group is (up to isomorphism) symmetric group:S5 (see subgroup structure of symmetric group:S5).
VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

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 $H = H_{4, 5}$ as the subgroup fixing ${4, 5}$ , so $H$ is symmetric group:S3 acting on the set ${1, 2, 3}$ .

$H$ has nine other conjugate subgroups, each corresponding to a different pair of fixed points:

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

Arithmetic functions

Function	Value	Explanation
order of group	120
order of subgroup	6
index of subgroup	20
size of conjugacy class of subgroup	10	conjugacy class is made of those listed above

Subgroup properties

Other properties

Property	Meaning	Satisfied?	Explanation	Comment
Hall subgroup	order and index are relatively prime	No	Order (6) and index (20) are not relatively prime
normal subgroup		No

GAP implementation

The group-subgroup pair can be constructed as follows:

G := SymmetricGroup(5); H := SymmetricGroup(3);