D₁₃₅₃₆ #72235533

order	27072
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₅₃₆

Splittings

Fitting	C₉×C₃₂×C₄₇ ⋊ C₂
Hall	C₉×C₄₇ ⋊ D₃₂
Sylow	C₄₇ ⋊ D₂₈₈
Sylow	C₉ ⋊ D₁₅₀₄

Subgroups

	subgroup	G/H
center	C₂	D₆₇₆₈
commutator, cosocle	C₉×C₁₆×C₄₇	C₂²
Frattini	C₃×C₁₆	C₂×D₁₄₁
Fitting	C₉×C₃₂×C₄₇	C₂
socle	C₂×C₃×C₄₇	D₄₈
hypercenter	C₃₂	D₄₂₃
hypocenter	C₉×C₄₇	D₃₂
LCS	G ▷ C₉×C₁₆×C₄₇ ▷ C₈×C₉×C₄₇ ▷ C₄×C₉×C₄₇ ▷ C₂×C₉×C₄₇ ▷ C₉×C₄₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 88
transitive degree	μ_t(G) = 13536

A minimal faithful permutation representation — 2 generators on 88 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)(42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88);(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)(34,41)(35,40)(36,39)(37,38)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)(64,67)(65,66)

Polycyclic presentation:

2^6,3,47,3|b^2=cd,c^2=de,d^2=ef,e^2=f,g^3=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2i^2,h^a=h^46,i^a=i^2

Conjugacy classes & character table

6771 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₇₀₇₂
maximal normal subgroup of	D₂₇₀₇₂
maximal quotient of	Dic₁₃₅₃₆, D₂₇₀₇₂

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 47, 3]);;
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> SetPower(coll, 7, F.9);;
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.7^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^46);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13536 #72235533