D₁₂₁₈₉ #37567158

order	24378
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₁₈₉
unambiguous semidirect product	C₃⋊D₄₀₆₃
unambiguous semidirect product	C₁₇⋊D₇₁₇
unambiguous semidirect product	C₂₃₉⋊D₅₁

Splittings

Fitting, socle, derived, Hall	C₃×C₁₇×C₂₃₉ ⋊ C₂
Hall	C₁₇×C₂₃₉ ⋊ D₃
Hall	C₃×C₂₃₉ ⋊ D₁₇
Sylow	C₂₃₉ ⋊ D₅₁
Hall	C₃×C₁₇ ⋊ D₂₃₉
Sylow	C₁₇ ⋊ D₇₁₇
Sylow	C₃ ⋊ D₄₀₆₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₇×C₂₃₉	C₂
LCS	G ▷ C₃×C₁₇×C₂₃₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 259
transitive degree	μ_t(G) = 12189

A minimal faithful permutation representation — 2 generators on 259 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259);(2,3)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(22,259)(23,258)(24,257)(25,256)(26,255)(27,254)(28,253)(29,252)(30,251)(31,250)(32,249)(33,248)(34,247)(35,246)(36,245)(37,244)(38,243)(39,242)(40,241)(41,240)(42,239)(43,238)(44,237)(45,236)(46,235)(47,234)(48,233)(49,232)(50,231)(51,230)(52,229)(53,228)(54,227)(55,226)(56,225)(57,224)(58,223)(59,222)(60,221)(61,220)(62,219)(63,218)(64,217)(65,216)(66,215)(67,214)(68,213)(69,212)(70,211)(71,210)(72,209)(73,208)(74,207)(75,206)(76,205)(77,204)(78,203)(79,202)(80,201)(81,200)(82,199)(83,198)(84,197)(85,196)(86,195)(87,194)(88,193)(89,192)(90,191)(91,190)(92,189)(93,188)(94,187)(95,186)(96,185)(97,184)(98,183)(99,182)(100,181)(101,180)(102,179)(103,178)(104,177)(105,176)(106,175)(107,174)(108,173)(109,172)(110,171)(111,170)(112,169)(113,168)(114,167)(115,166)(116,165)(117,164)(118,163)(119,162)(120,161)(121,160)(122,159)(123,158)(124,157)(125,156)(126,155)(127,154)(128,153)(129,152)(130,151)(131,150)(132,149)(133,148)(134,147)(135,146)(136,145)(137,144)(138,143)(139,142)(140,141)

Polycyclic presentation:

2,3,17,239|b^a=b^2,c^a=c^16,d^a=d^238

Conjugacy classes & character table

6096 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24378, 9);

D12189 #37567158