C₅×D₃×C₂₂₃⋊C₃ #34095475

order	20070
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₃×C₂₂₃⋊C₃
unambiguous semidirect product	C₃⋊(C₂×C₅×C₂₂₃⋊C₃)
unambiguous semidirect product	(D₃×C₂₂₃)⋊(C₃×C₅)
unambiguous semidirect product	(C₅×D₃×C₂₂₃)⋊C₃

Splittings

Fitting, socle	C₃×C₅×C₂₂₃ ⋊ C₂×C₃
derived	C₃×C₂₂₃ ⋊ C₂×C₃×C₅
Hall	C₃×C₅×C₂₂₃⋊C₃ ⋊ C₂
Hall	C₃×C₂₂₃⋊C₃ ⋊ C₂×C₅
Hall	C₅×C₂₂₃ ⋊ C₃×D₃
Sylow	C₂₂₃ ⋊ C₃×C₅×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₃×C₂₂₃⋊C₃
commutator, cosocle, hypocenter	C₃×C₂₂₃	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₂₂₃	C₂×C₃
LCS	G ▷ C₃×C₂₂₃
UCS	C₁ ◁ C₅

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 231
transitive degree	μ_t(G) = 3345

A minimal faithful permutation representation — 2 generators on 231 points:

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

Conjugacy classes & character table

1155 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20070, 18);

C5×D3×C223⋊C3 #34095475