A5 in A6

This article describes a subgroup $$H$$ in a group $$G$$, where $$G$$ is alternating group:A6, the alternating group on the set $$\{ 1,2,3,4,5,6\}$$, and $$H = H_6$$ is the subgroup of $$G$$ comprising those permutations that fix $$\{ 6 \}$$. $$H$$ can be identified with alternating group:A5 via its action on $$\{ 1,2,3,4,5\}$$.

$$H$$ has five other conjugate subgroups:


 * $$H_1$$ is the subgroup of $$G$$ fixing $$\{ 1 \}$$
 * $$H_2$$ is the subgroup of $$G$$ fixing $$\{ 2 \}$$
 * $$H_3$$ is the subgroup of $$G$$ fixing $$\{ 3 \}$$
 * $$H_4$$ is the subgroup of $$G$$ fixing $$\{ 4 \}$$
 * $$H_5$$ is the subgroup of $$G$$ fixing $$\{ 5 \}$$

Construction of group-subgroup pair
The group-subgroup pair can be constructed using the AlternatingGroup function:

G := AlternatingGroup(6); H := AlternatingGroup(5);