D₁₆₉₀₅ #46798913

order	33810
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₉₀₅
unambiguous semidirect product	C₃⋊D₅₆₃₅
unambiguous semidirect product	C₅⋊D₃₃₈₁
unambiguous semidirect product	C₂₃⋊D₇₃₅
unambiguous semidirect product	C₄₉⋊D₃₄₅

Splittings

Fitting, derived, Hall	C₃×C₅×C₂₃×C₄₉ ⋊ C₂
Hall	C₅×C₂₃×C₄₉ ⋊ D₃
Hall	C₃×C₂₃×C₄₉ ⋊ D₅
Hall	C₂₃×C₄₉ ⋊ D₁₅
Hall	C₃×C₅×C₄₉ ⋊ D₂₃
Hall	C₃×C₅×C₂₃ ⋊ D₄₉
Hall	C₅×C₄₉ ⋊ D₆₉
Hall	C₃×C₄₉ ⋊ D₁₁₅
Hall	C₅×C₂₃ ⋊ D₁₄₇
Hall	C₃×C₂₃ ⋊ D₂₄₅
Sylow	C₄₉ ⋊ D₃₄₅
Sylow	C₂₃ ⋊ D₇₃₅
Hall	C₃×C₅ ⋊ D₁₁₂₇
Sylow	C₅ ⋊ D₃₃₈₁
Sylow	C₃ ⋊ D₅₆₃₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₅×C₂₃×C₄₉	C₂
Frattini	C₇	D₂₄₁₅
socle	C₃×C₅×C₇×C₂₃	D₇
LCS	G ▷ C₃×C₅×C₂₃×C₄₉
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 16905

A minimal faithful permutation representation — 2 generators on 80 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);(2,3)(5,8)(6,7)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

2,3,5,7,23,7|d^7=f^6,b^a=b^2,c^a=c^4,d^a=d^6f,e^a=e^22,f^a=f^6

Conjugacy classes & character table

8454 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33810, 17);

D16905 #46798913