# A5 in A6

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) alternating group:A5 and the group is (up to isomorphism) alternating group:A6 (see subgroup structure of alternating group:A6).
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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 $\{ 5 \}$

## Arithmetic functions

Function Value Explanation Comment
order of the group 360
order of the subgroup 60
index of the subgroup 6
size of conjugacy class 6
number of conjugacy classes in automorphism class 2

## GAP implementation

### Construction of group-subgroup pair

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

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