D₁₂₀₄₅ #37562799

order	24090
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₀₄₅
unambiguous semidirect product	C₃⋊D₄₀₁₅
unambiguous semidirect product	C₅⋊D₂₄₀₉
unambiguous semidirect product	C₁₁⋊D₁₀₉₅
unambiguous semidirect product	C₇₃⋊D₁₆₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₁₁×C₇₃ ⋊ C₂
Hall	C₅×C₁₁×C₇₃ ⋊ D₃
Hall	C₃×C₁₁×C₇₃ ⋊ D₅
Hall	C₃×C₅×C₇₃ ⋊ D₁₁
Hall	C₁₁×C₇₃ ⋊ D₁₅
Hall	C₅×C₇₃ ⋊ D₃₃
Hall	C₃×C₇₃ ⋊ D₅₅
Hall	C₃×C₅×C₁₁ ⋊ D₇₃
Sylow	C₇₃ ⋊ D₁₆₅
Hall	C₅×C₁₁ ⋊ D₂₁₉
Hall	C₃×C₁₁ ⋊ D₃₆₅
Hall	C₃×C₅ ⋊ D₈₀₃
Sylow	C₁₁ ⋊ D₁₀₉₅
Sylow	C₅ ⋊ D₂₄₀₉
Sylow	C₃ ⋊ D₄₀₁₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₁₁×C₇₃	C₂
LCS	G ▷ C₃×C₅×C₁₁×C₇₃
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 92
transitive degree	μ_t(G) = 12045

A minimal faithful permutation representation — 2 generators on 92 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92);(2,3)(5,8)(6,7)(10,19)(11,18)(12,17)(13,16)(14,15)(21,92)(22,91)(23,90)(24,89)(25,88)(26,87)(27,86)(28,85)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)

Polycyclic presentation:

2,3,5,11,73|b^a=b^2,c^a=c^4,d^a=d^10,e^a=e^72

Conjugacy classes & character table

6024 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24090, 19);

D12045 #37562799