S4 in S5

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 $$\{ 1,2,3,4 \}$$.

GAP implementation
The group-subgroup pair can be constructed as follows:

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