D₁₈₁₆₅ #49138102

order	36330
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₈₁₆₅
unambiguous semidirect product	C₃⋊D₆₀₅₅
unambiguous semidirect product	C₅⋊D₃₆₃₃
unambiguous semidirect product	C₇⋊D₂₅₉₅
unambiguous semidirect product	C₁₇₃⋊D₁₀₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₇×C₁₇₃ ⋊ C₂
Hall	C₅×C₇×C₁₇₃ ⋊ D₃
Hall	C₃×C₇×C₁₇₃ ⋊ D₅
Hall	C₃×C₅×C₁₇₃ ⋊ D₇
Hall	C₇×C₁₇₃ ⋊ D₁₅
Hall	C₅×C₁₇₃ ⋊ D₂₁
Hall	C₃×C₁₇₃ ⋊ D₃₅
Sylow	C₁₇₃ ⋊ D₁₀₅
Hall	C₃×C₅×C₇ ⋊ D₁₇₃
Hall	C₅×C₇ ⋊ D₅₁₉
Hall	C₃×C₇ ⋊ D₈₆₅
Hall	C₃×C₅ ⋊ D₁₂₁₁
Sylow	C₇ ⋊ D₂₅₉₅
Sylow	C₅ ⋊ D₃₆₃₃
Sylow	C₃ ⋊ D₆₀₅₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₇×C₁₇₃	C₂
LCS	G ▷ C₃×C₅×C₇×C₁₇₃
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 188
transitive degree	μ_t(G) = 18165

A minimal faithful permutation representation — 2 generators on 188 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188);(2,3)(5,8)(6,7)(10,15)(11,14)(12,13)(17,188)(18,187)(19,186)(20,185)(21,184)(22,183)(23,182)(24,181)(25,180)(26,179)(27,178)(28,177)(29,176)(30,175)(31,174)(32,173)(33,172)(34,171)(35,170)(36,169)(37,168)(38,167)(39,166)(40,165)(41,164)(42,163)(43,162)(44,161)(45,160)(46,159)(47,158)(48,157)(49,156)(50,155)(51,154)(52,153)(53,152)(54,151)(55,150)(56,149)(57,148)(58,147)(59,146)(60,145)(61,144)(62,143)(63,142)(64,141)(65,140)(66,139)(67,138)(68,137)(69,136)(70,135)(71,134)(72,133)(73,132)(74,131)(75,130)(76,129)(77,128)(78,127)(79,126)(80,125)(81,124)(82,123)(83,122)(84,121)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)

Polycyclic presentation:

2,3,5,7,173|b^a=b^2,c^a=c^4,d^a=d^6,e^a=e^172

Conjugacy classes & character table

9084 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36330, 17);

D18165 #49138102