D₆₈₆₄ #72212434

order	13728
automorphism	Aut(G) order 13178880
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₆₈₆₄

Splittings

Fitting	C₃×C₁₁×C₁₃×C₁₆ ⋊ C₂
Hall	C₃×C₁₁×C₁₃ ⋊ D₁₆
Hall	C₁₁×C₁₃ ⋊ D₄₈
Hall	C₃×C₁₃ ⋊ D₁₇₆
Hall	C₃×C₁₁ ⋊ D₂₀₈
Sylow	C₁₃ ⋊ D₅₂₈
Sylow	C₁₁ ⋊ D₆₂₄
Sylow	C₃ ⋊ D₂₂₈₈

Subgroups

	subgroup	G/H
center	C₂	D₃₄₃₂
commutator, cosocle	C₃×C₈×C₁₁×C₁₃	C₂²
Frattini	C₈	C₂×D₄₂₉
Fitting	C₃×C₁₁×C₁₃×C₁₆	C₂
socle	C₂×C₃×C₁₁×C₁₃	D₈
hypercenter	C₁₆	D₄₂₉
hypocenter	C₃×C₁₁×C₁₃	D₁₆
LCS	G ▷ C₃×C₈×C₁₁×C₁₃ ▷ C₃×C₄×C₁₁×C₁₃ ▷ C₂×C₃×C₁₁×C₁₃ ▷ C₃×C₁₁×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 6864

A minimal faithful permutation representation — 2 generators on 43 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19)(20,21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,30)(22,29)(23,28)(24,27)(25,26)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)

Polycyclic presentation:

2^5,3,11,13|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^10,h^a=h^12

Conjugacy classes & character table

3435 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₃₇₂₈, D₂₀₅₉₂
maximal normal subgroup of	D₁₃₇₂₈
maximal quotient of	D₁₃₇₂₈, Dic₆₈₆₄, D₂₀₅₉₂

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 11, 13]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 8, 1, F.8^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D6864 #72212434