D₂₉₉₂₀ #72294993

order	59840
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₉₉₂₀

Splittings

Fitting	C₅×C₁₁×C₁₇×C₃₂ ⋊ C₂
Hall	C₅×C₁₁×C₁₇ ⋊ D₃₂
Hall	C₁₁×C₁₇ ⋊ D₁₆₀
Hall	C₅×C₁₇ ⋊ D₃₅₂
Hall	C₅×C₁₁ ⋊ D₅₄₄
Sylow	C₁₇ ⋊ D₁₇₆₀
Sylow	C₁₁ ⋊ D₂₇₂₀
Sylow	C₅ ⋊ D₅₉₈₄

Subgroups

	subgroup	G/H
center	C₂	D₁₄₉₆₀
commutator, cosocle	C₅×C₁₁×C₁₆×C₁₇	C₂²
Frattini	C₁₆	C₂×D₉₃₅
Fitting	C₅×C₁₁×C₁₇×C₃₂	C₂
socle	C₂×C₅×C₁₁×C₁₇	D₁₆
hypercenter	C₃₂	D₉₃₅
hypocenter	C₅×C₁₁×C₁₇	D₃₂
LCS	G ▷ C₅×C₁₁×C₁₆×C₁₇ ▷ C₅×C₈×C₁₁×C₁₇ ▷ C₄×C₅×C₁₁×C₁₇ ▷ C₂×C₅×C₁₁×C₁₇ ▷ C₅×C₁₁×C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 29920

A minimal faithful permutation representation — 2 generators on 65 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);(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)(34,37)(35,36)(39,48)(40,47)(41,46)(42,45)(43,44)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Polycyclic presentation:

2^6,5,11,17|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^4,h^a=h^10,i^a=i^16

Conjugacy classes & character table

14963 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, 5, 11, 17]);;
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.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^10);;
gap> SetConjugate(coll, 9, 1, F.9^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D29920 #72294993