D8 in S4

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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) dihedral group:D8 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).
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This article is about the subgroup H in the group G, where G is symmetric group:S4, i.e., the symmetric group on the set \{ 1,2,3,4 \}, and H is the subgroup:

\! \{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2), (1,2)(3,4), (1,4)(2,3), (1,3), (2,4) \}

H is a 2-Sylow subgroup of G, and has two other conjugate subgroups, which are given below:

\! H_1 = \{ (), (1,3,2,4), (1,2)(3,4), (1,4,2,3), (1,3)(2,4), (1,4)(2,3), (1,2), (3,4) \}

and:

\! H_2 = \{ (), (1,2,4,3), (1,4)(2,3), (1,3,4,2), (1,2)(3,4), (1,3)(2,4), (1,4), (2,3) \}