Z5 in A5

Definition
The group $$G$$ is alternating group:A5, which for concreteness, we take as the alternating group on the set $$\{ 1,2,3,4, 5 \}$$.

$$H$$ is the subgroup:

$$\! H := \{, (1,2,3,4,5), (1,3,5,2,4), (1,4,2,5,3), (1,5,4,3,2) \}$$

There is a total of five other conjugate subgroups (so six subgroups, including $$H$$). The other subgroups are:


 * $$\! \{, (1,2,3,5,4), (1,3,4,2,5), (1,5,2,4,3), (1,4,5,3,2) \}$$
 * $$\! \{, (1,2,4,3,5), (1,4,5,2,3), (1,3,2,5,4), (1,5,3,4,2) \}$$
 * $$\! \{, (1,2,4,5,3), (1,4,3,2,5), (1,5,2,3,4), (1,3,5,4,2) \}$$
 * $$\! \{, (1,2,5,3,4), (1,5,4,2,3), (1,3,2,4,5), (1,4,3,5,2) \}$$
 * $$\! \{, (1,2,5,4,3), (1,5,3,2,4), (1,4,2,3,5), (1,3,4,5,2) \}$$

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

G := AlternatingGroup(5); H := Group([(1,2,3,4,5)]);