D₃₂₀₃₂ #72301879

order	64064
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₀₃₂

Splittings

Fitting	C₇×C₁₁×C₁₃×C₃₂ ⋊ C₂
Hall	C₇×C₁₁×C₁₃ ⋊ D₃₂
Hall	C₁₁×C₁₃ ⋊ D₂₂₄
Hall	C₇×C₁₃ ⋊ D₃₅₂
Hall	C₇×C₁₁ ⋊ D₄₁₆
Sylow	C₁₃ ⋊ D₂₄₆₄
Sylow	C₁₁ ⋊ D₂₉₁₂
Sylow	C₇ ⋊ D₄₅₇₆

Subgroups

	subgroup	G/H
center	C₂	D₁₆₀₁₆
commutator, cosocle	C₇×C₁₁×C₁₃×C₁₆	C₂²
Frattini	C₁₆	C₂×D₁₀₀₁
Fitting	C₇×C₁₁×C₁₃×C₃₂	C₂
socle	C₂×C₇×C₁₁×C₁₃	D₁₆
hypercenter	C₃₂	D₁₀₀₁
hypocenter	C₇×C₁₁×C₁₃	D₃₂
LCS	G ▷ C₇×C₁₁×C₁₃×C₁₆ ▷ C₇×C₈×C₁₁×C₁₃ ▷ C₄×C₇×C₁₁×C₁₃ ▷ C₂×C₇×C₁₁×C₁₃ ▷ C₇×C₁₁×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 32032

A minimal faithful permutation representation — 2 generators on 63 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);(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)(34,39)(35,38)(36,37)(41,50)(42,49)(43,48)(44,47)(45,46)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Polycyclic presentation:

2^6,7,11,13|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^6,h^a=h^10,i^a=i^12

Conjugacy classes & character table

16019 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, 7, 11, 13]);;
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.7^6);;
gap> SetConjugate(coll, 8, 1, F.8^10);;
gap> SetConjugate(coll, 9, 1, F.9^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32032 #72301879