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:

\! H = \{ (), (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 is isomorphic to dihedral group:D8. It 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) \}

Complements

The permutable complements to H (and also to each of its conjugates) are precisely the subgroups of order three, namely A3 in S4 and its conjugates. Each of these is a conjugate to each of the conjugates of H:

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

In addition, any S2 in S4 is a lattice complement but not a permutable complement to two of the three conjugates of H, namely the ones that do not contain it.

Properties related to complementation

Property Meaning Satisfied? Explanation Comment
permutably complemented subgroup has a permutable complement Yes A3 in S4
lattice-complemented subgroup has a lattice complement Yes A3 in S4, or S2 in S4
retract has a normal complement No
complemented normal subgroup No not a normal subgroup

Arithmetic functions

Function Value Explanation Comment
order of the group 24
order of the subgroup 8
index of the subgroup 3
size of conjugacy class 3
number of conjugacy classes in automorphism class 1

Effect of subgroup operators

In the table below, we provide values specific to H.

Function Value as subgroup (descriptive) Value as subgroup (link) Value as group
normalizer the subgroup itself (current page) dihedral group:D8
centralizer \{ (), (1,3)(2,4) \} subgroup generated by double transposition in S4 cyclic group:Z2
normal core \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \} normal Klein four-subgroup of symmetric group:S4 Klein four-group
normal closure the whole group -- symmetric group:S4
characteristic core \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \} normal Klein four-subgroup of symmetric group:S4 Klein four-group
characteristic closure the whole group -- symmetric group:S4

Subgroup properties

Resemblance-based properties

Property Meaning Satisfied? Explanation Comment
order-conjugate subgroup conjugate to any subgroup of the same order Yes Sylow implies order-conjugate
order-dominating subgroup any subgroup of the whole group whose order divides the order of H is contained in a conjugate of H Yes Sylow implies order-dominating
order-dominated subgroup any subgroup of the whole group whose order is a multiple of the order of H contains a conjugate of H Yes Sylow implies order-dominated
order-isomorphic subgroup isomorphic to any subgroup of the group of the same order Yes (via order-conjugate)
isomorph-automorphic subgroup Yes (via order-conjugate)
automorph-conjugate subgroup Yes (via order-conjugate)
order-automorphic subgroup Yes (via order-conjugate)
isomorph-conjugate subgroup Yes (via order-conjugate)

Normality-related properties

Property Meaning Satisfied? Explanation Comment
normal subgroup equals all its conjugate subgroups No (see other conjugate subgroups)
subnormal subgroup No
self-normalizing subgroup equals its normalizer in the whole group Yes
abnormal subgroup Yes
weakly abnormal subgroup Yes
contranormal subgroup Yes
maximal subgroup Yes
pronormal subgroup Yes Sylow implies pronormal
weakly pronormal subgroup Yes (via pronormal)

Fusion system on subgroup

The subgroup embedding induces the non-inner non-simple fusion system for dihedral group:D8.

GAP implementation

Construction of subgroup given group as a black box

Suppose we are already given a group G that we know to be isomorphic to symmetric group:S4. Then, the subgroup H can be constructed using SylowSubgroup as follows:

H := SylowSubgroup(G,2);

Construction of group-subgroup pair

The group and subgroup pair can be defined using GAP's SymmetricGroup and SylowSubgroup functions as follows:

G := SymmetricGroup(4); H := SylowSubgroup(G,2);

Note that this doesn't output the subgroup H used on this page, but rather the conjugate subgroup H_1. However all relevant properties are invariant under conjugation, so this does not matter.