D₁₁₇₁₅ #37552873

order	23430
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₇₁₅
unambiguous semidirect product	C₃⋊D₃₉₀₅
unambiguous semidirect product	C₅⋊D₂₃₄₃
unambiguous semidirect product	C₁₁⋊D₁₀₆₅
unambiguous semidirect product	C₇₁⋊D₁₆₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₁₁×C₇₁ ⋊ C₂
Hall	C₅×C₁₁×C₇₁ ⋊ D₃
Hall	C₃×C₁₁×C₇₁ ⋊ D₅
Hall	C₃×C₅×C₇₁ ⋊ D₁₁
Hall	C₁₁×C₇₁ ⋊ D₁₅
Hall	C₅×C₇₁ ⋊ D₃₃
Hall	C₃×C₇₁ ⋊ D₅₅
Hall	C₃×C₅×C₁₁ ⋊ D₇₁
Sylow	C₇₁ ⋊ D₁₆₅
Hall	C₅×C₁₁ ⋊ D₂₁₃
Hall	C₃×C₁₁ ⋊ D₃₅₅
Hall	C₃×C₅ ⋊ D₇₈₁
Sylow	C₁₁ ⋊ D₁₀₆₅
Sylow	C₅ ⋊ D₂₃₄₃
Sylow	C₃ ⋊ D₃₉₀₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₁₁×C₇₁	C₂
LCS	G ▷ C₃×C₅×C₁₁×C₇₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 90
transitive degree	μ_t(G) = 11715

A minimal faithful permutation representation — 2 generators on 90 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90);(2,3)(5,8)(6,7)(10,19)(11,18)(12,17)(13,16)(14,15)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2,3,5,11,71|b^a=b^2,c^a=c^4,d^a=d^10,e^a=e^70

Conjugacy classes & character table

5859 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23430, 22);

D11715 #37552873