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
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	Comment
Hall subgroup	order and index are relatively prime	Yes	$\{ 2,3 \}$-Hall subgroup. Also, order (24) and index (5) are relatively prime.
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);