C₃²×D₁₁₁₅ #34095481

order	20070
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃²×D₁₁₁₅
unambiguous semidirect product	C₅⋊(C₃²×D₂₂₃)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₂₂₃ ⋊ C₂
derived, Hall	C₅×C₂₂₃ ⋊ C₂×C₃²
Hall	C₃²×C₂₂₃ ⋊ D₅
Sylow	C₂₂₃ ⋊ C₃²×D₅
Hall	C₃²×C₅ ⋊ D₂₂₃
Sylow	C₅ ⋊ C₃²×D₂₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	D₁₁₁₅
commutator, cosocle, hypocenter	C₅×C₂₂₃	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₂₂₃	C₂
LCS	G ▷ C₅×C₂₂₃
UCS	C₁ ◁ C₃²

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 234
transitive degree	μ_t(G) = 10035

A minimal faithful permutation representation — 2 generators on 234 points:

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

Conjugacy classes & character table

5031 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20070, 24);

C32×D1115 #34095481