D₃₂₇₃₆ #72304356

order	65472
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₇₃₆

Splittings

Fitting	C₃×C₁₁×C₃₁×C₃₂ ⋊ C₂
Hall	C₃×C₁₁×C₃₁ ⋊ D₃₂
Hall	C₁₁×C₃₁ ⋊ D₉₆
Hall	C₃×C₃₁ ⋊ D₃₅₂
Hall	C₃×C₁₁ ⋊ D₉₉₂
Sylow	C₃₁ ⋊ D₁₀₅₆
Sylow	C₁₁ ⋊ D₂₉₇₆
Sylow	C₃ ⋊ D₁₀₉₁₂

Subgroups

	subgroup	G/H
center	C₂	D₁₆₃₆₈
commutator, cosocle	C₃×C₁₁×C₁₆×C₃₁	C₂²
Frattini	C₁₆	C₂×D₁₀₂₃
Fitting	C₃×C₁₁×C₃₁×C₃₂	C₂
socle	C₂×C₃×C₁₁×C₃₁	D₁₆
hypercenter	C₃₂	D₁₀₂₃
hypocenter	C₃×C₁₁×C₃₁	D₃₂
LCS	G ▷ C₃×C₁₁×C₁₆×C₃₁ ▷ C₃×C₈×C₁₁×C₃₁ ▷ C₃×C₄×C₁₁×C₃₁ ▷ C₂×C₃×C₁₁×C₃₁ ▷ C₃×C₁₁×C₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 32736

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

Polycyclic presentation:

2^6,3,11,31|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^2,h^a=h^10,i^a=i^30

Conjugacy classes & character table

16371 conjugacy classes. The character table is too large to display.

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, 11, 31]);;
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.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^10);;
gap> SetConjugate(coll, 9, 1, F.9^30);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32736 #72304356