D₁₁₆₆₁ #37552218

order	23322
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₆₆₁
unambiguous semidirect product	C₃⋊D₃₈₈₇
unambiguous semidirect product	C₂₃⋊D₅₀₇
unambiguous semidirect product	C₁₆₉⋊D₆₉

Splittings

Fitting, derived, Hall	C₃×C₂₃×C₁₆₉ ⋊ C₂
Hall	C₂₃×C₁₆₉ ⋊ D₃
Hall	C₃×C₁₆₉ ⋊ D₂₃
Sylow	C₁₆₉ ⋊ D₆₉
Hall	C₃×C₂₃ ⋊ D₁₆₉
Sylow	C₂₃ ⋊ D₅₀₇
Sylow	C₃ ⋊ D₃₈₈₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃×C₂₃×C₁₆₉	C₂
Frattini	C₁₃	D₈₉₇
socle	C₃×C₁₃×C₂₃	D₁₃
LCS	G ▷ C₃×C₂₃×C₁₆₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 195
transitive degree	μ_t(G) = 11661

A minimal faithful permutation representation — 2 generators on 195 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)(173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195);(2,3)(5,172)(6,171)(7,170)(8,169)(9,168)(10,167)(11,166)(12,165)(13,164)(14,163)(15,162)(16,161)(17,160)(18,159)(19,158)(20,157)(21,156)(22,155)(23,154)(24,153)(25,152)(26,151)(27,150)(28,149)(29,148)(30,147)(31,146)(32,145)(33,144)(34,143)(35,142)(36,141)(37,140)(38,139)(39,138)(40,137)(41,136)(42,135)(43,134)(44,133)(45,132)(46,131)(47,130)(48,129)(49,128)(50,127)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(174,195)(175,194)(176,193)(177,192)(178,191)(179,190)(180,189)(181,188)(182,187)(183,186)(184,185)

Polycyclic presentation:

2,3,13,23,13|c^13=e^12,b^a=b^2,c^a=c^12e,d^a=d^22,e^a=e^12

Conjugacy classes & character table

5832 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23322, 12);

D11661 #37552218