C₃×C₅⋊(D₂₂₃⋊C₃) #34095472

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

Aliases

direct product	C₃×C₅⋊(D₂₂₃⋊C₃)
unambiguous semidirect product	C₅⋊(C₃×D₂₂₃⋊C₃)
unambiguous semidirect product	(C₂₂₃⋊C₃)⋊(C₃×D₅)
unambiguous semidirect product	D₁₁₁₅⋊C₃²
unambiguous semidirect product	(C₃×D₁₁₁₅)⋊C₃

Splittings

Fitting, socle	C₃×C₅×C₂₂₃ ⋊ C₂×C₃
derived, Hall	C₅×C₂₂₃ ⋊ C₂×C₃²
Hall	C₃×C₅×C₂₂₃⋊C₃ ⋊ C₂
Hall	D₁₁₁₅ ⋊ C₃²
Hall	C₃×C₂₂₃⋊C₃ ⋊ D₅
Sylow	C₂₂₃ ⋊ C₃²×D₅
Sylow	C₅ ⋊ C₃×D₂₂₃⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₅⋊(D₂₂₃⋊C₃)
commutator, cosocle, hypocenter	C₅×C₂₂₃	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₂₂₃	C₂×C₃
LCS	G ▷ C₅×C₂₂₃
UCS	C₁ ◁ C₃

48 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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20070, 15);

C3×C5⋊(D223⋊C3) #34095472