C₅×D₉×D₁₆₉ #44477448

order	30420
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₉×D₁₆₉
unambiguous semidirect product	(D₉×C₁₆₉)⋊(C₂×C₅)
unambiguous semidirect product	(C₉×D₁₆₉)⋊(C₂×C₅)
unambiguous semidirect product	D₁₅₂₁⋊(C₂×C₅)

Splittings

Fitting, Hall	C₅×C₉×C₁₆₉ ⋊ C₂²
derived, Hall	C₉×C₁₆₉ ⋊ C₂²×C₅
Hall	C₅×C₁₆₉ ⋊ C₂×D₉
Sylow	C₁₆₉ ⋊ C₂×C₅×D₉
Hall	C₅×C₉ ⋊ C₂×D₁₆₉
Sylow	C₉ ⋊ C₂×C₅×D₁₆₉

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₉×D₁₆₉
commutator, cosocle, hypocenter	C₉×C₁₆₉	C₂²×C₅
Frattini	C₃×C₁₃	C₅×D₃×D₁₃
Fitting	C₅×C₉×C₁₆₉	C₂²
socle	C₃×C₅×C₁₃	D₃×D₁₃
LCS	G ▷ C₉×C₁₆₉
UCS	C₁ ◁ C₅

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 183
transitive degree	μ_t(G) = 7605

A minimal faithful permutation representation — 2 generators on 183 points:

(1,2,3,4,5)(6,11,9,8,13,10,12,7,14)(16,18)(17,21)(19,24)(20,27)(22,30)(23,29)(25,34)(26,37)(28,39)(31,43)(32,42)(33,41)(35,48)(36,51)(38,53)(40,55)(44,60)(45,59)(46,58)(47,57)(49,66)(50,69)(52,71)(54,73)(56,75)(61,81)(62,80)(63,79)(64,78)(65,77)(67,88)(68,91)(70,93)(72,95)(74,97)(76,99)(82,106)(83,105)(84,104)(85,103)(86,102)(87,101)(89,112)(90,114)(92,116)(94,118)(96,120)(98,122)(100,124)(107,125)(108,129)(109,128)(110,127)(111,126)(113,134)(115,136)(117,138)(119,140)(121,142)(123,144)(130,145)(131,148)(132,147)(133,146)(135,152)(137,154)(139,156)(141,158)(143,160)(149,161)(150,163)(151,162)(153,166)(155,168)(157,170)(159,172)(164,173)(165,174)(167,176)(169,178)(171,180)(175,181)(177,182)(179,183);(7,9)(8,12)(10,13)(11,14)(15,76,79,52,173,101,155,154,111,172,51,84,75,17,100,58,72,163,49,169,136,133,158,34,108,55,22,125,41,96,147,68,179,114,151,140,47,130,39,31,105,19,121,127,92,182,88,165,118,64,144,27,44,80,26,143,102,117,176,89,175,93,85,122,18,61,59,38,161,77,139,166,113,180,69,109,97,23,82,42,54,148,35,157,152,135,170,48,131,73,32,106,29,74,128,50,171,134,153,156,65,149,53,45,81,16,98,103,70,181,112,167,138,86,160,37,62,60,20,123,78,94,174,67,177,116,110,142,24,83,43,28,145,57,119,162,90,183,91,132,120,33,107,30,40,129,25,141,146,115,178,66,150,95,46,124,21,56,104,36,159,126,137,168,87,164,71,63,99)

Conjugacy classes & character table

2580 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 37);

C5×D9×D169 #44477448