C₁₉×C₂₁₁⋊C₅ #34095302

order	20045
automorphism	?
homology	Abelianization (H₁) = C₅×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×C₂₁₁⋊C₅
unambiguous semidirect product	C₂₁₁⋊(C₅×C₁₉)
unambiguous semidirect product	(C₁₉×C₂₁₁)⋊C₅

Splittings

Fitting, socle, Hall	C₁₉×C₂₁₁ ⋊ C₅
derived, Sylow	C₂₁₁ ⋊ C₅×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉	C₂₁₁⋊C₅
commutator, cosocle, hypocenter	C₂₁₁	C₅×C₁₉
Frattini	C₁	G
Fitting, socle	C₁₉×C₂₁₁	C₅
LCS	G ▷ C₂₁₁
UCS	C₁ ◁ C₁₉

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 230
transitive degree	μ_t(G) = 4009

A minimal faithful permutation representation — 2 generators on 230 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20045, 1);

C19×C211⋊C5 #34095302