D₁₄₄₃₂ #72235781

order	28864
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₄₄₃₂

Splittings

Fitting	C₁₁×C₃₂×C₄₁ ⋊ C₂
Hall	C₁₁×C₄₁ ⋊ D₃₂
Sylow	C₄₁ ⋊ D₃₅₂
Sylow	C₁₁ ⋊ D₁₃₁₂

Subgroups

	subgroup	G/H
center	C₂	D₇₂₁₆
commutator, cosocle	C₁₁×C₁₆×C₄₁	C₂²
Frattini	C₁₆	C₂×D₄₅₁
Fitting	C₁₁×C₃₂×C₄₁	C₂
socle	C₂×C₁₁×C₄₁	D₁₆
hypercenter	C₃₂	D₄₅₁
hypocenter	C₁₁×C₄₁	D₃₂
LCS	G ▷ C₁₁×C₁₆×C₄₁ ▷ C₈×C₁₁×C₄₁ ▷ C₄×C₁₁×C₄₁ ▷ C₂×C₁₁×C₄₁ ▷ C₁₁×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 14432

A minimal faithful permutation representation — 2 generators on 84 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)(44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,43)(35,42)(36,41)(37,40)(38,39)(45,84)(46,83)(47,82)(48,81)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(61,68)(62,67)(63,66)(64,65)

Polycyclic presentation:

2^6,11,41|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^10,h^a=h^40

Conjugacy classes & character table

7219 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₈₈₆₄
maximal normal subgroup of	D₂₈₈₆₄
maximal quotient of	Dic₁₄₄₃₂, D₂₈₈₆₄

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, 2, 11, 41]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
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, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 8, 1, F.8^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D14432 #72235781