D₁₁₉₉₉ #37562085

order	23998
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₉₉₉
unambiguous semidirect product	C₇₁⋊D₁₆₉
unambiguous semidirect product	C₁₆₉⋊D₇₁

Splittings

Fitting, derived, Hall	C₇₁×C₁₆₉ ⋊ C₂
Sylow	C₁₆₉ ⋊ D₇₁
Sylow	C₇₁ ⋊ D₁₆₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇₁×C₁₆₉	C₂
Frattini	C₁₃	D₉₂₃
socle	C₁₃×C₇₁	D₁₃
LCS	G ▷ C₇₁×C₁₆₉
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 240
transitive degree	μ_t(G) = 11999

A minimal faithful permutation representation — 2 generators on 240 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)(170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240);(2,169)(3,168)(4,167)(5,166)(6,165)(7,164)(8,163)(9,162)(10,161)(11,160)(12,159)(13,158)(14,157)(15,156)(16,155)(17,154)(18,153)(19,152)(20,151)(21,150)(22,149)(23,148)(24,147)(25,146)(26,145)(27,144)(28,143)(29,142)(30,141)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,130)(42,129)(43,128)(44,127)(45,126)(46,125)(47,124)(48,123)(49,122)(50,121)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(171,240)(172,239)(173,238)(174,237)(175,236)(176,235)(177,234)(178,233)(179,232)(180,231)(181,230)(182,229)(183,228)(184,227)(185,226)(186,225)(187,224)(188,223)(189,222)(190,221)(191,220)(192,219)(193,218)(194,217)(195,216)(196,215)(197,214)(198,213)(199,212)(200,211)(201,210)(202,209)(203,208)(204,207)(205,206)

Polycyclic presentation:

2,13,71,13|b^13=d^12,b^a=b^12d,c^a=c^70,d^a=d^12

Conjugacy classes & character table

6001 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23998, 4);

D11999 #37562085