A5 in A6

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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);