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);