C₅×C₃⋊(D₃₇₃⋊C₃) #46777067

order	33570
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₃⋊(D₃₇₃⋊C₃)
unambiguous semidirect product	C₃⋊(C₅×D₃₇₃⋊C₃)
unambiguous semidirect product	(C₃₇₃⋊C₃)⋊(C₅×D₃)
unambiguous semidirect product	D₁₁₁₉⋊(C₃×C₅)
unambiguous semidirect product	(C₅×D₁₁₁₉)⋊C₃

Splittings

Fitting, socle	C₃×C₅×C₃₇₃ ⋊ C₂×C₃
derived	C₃×C₃₇₃ ⋊ C₂×C₃×C₅
Hall	C₃×C₅×C₃₇₃⋊C₃ ⋊ C₂
Hall	C₃×C₃₇₃⋊C₃ ⋊ C₂×C₅
Hall	C₅×C₃₇₃ ⋊ C₃×D₃
Sylow	C₃₇₃ ⋊ C₃×C₅×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊(D₃₇₃⋊C₃)
commutator, cosocle, hypocenter	C₃×C₃₇₃	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₃₇₃	C₂×C₃
LCS	G ▷ C₃×C₃₇₃
UCS	C₁ ◁ C₅

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 381
transitive degree	μ_t(G) = 5595

A minimal faithful permutation representation — 2 generators on 381 points:

(1,2,3,4,5)(7,8)(10,14,15,11,12,16)(13,22,23,17,18,32)(19,35,36,25,26,49)(20,34,30,28,24,21)(27,54,55,38,39,72)(29,52,53,41,37,46)(31,33,51,43,44,47)(40,77,78,57,58,101)(42,75,76,60,56,69)(45,50,74,62,63,70)(48,90,88,66,67,65)(59,106,107,80,81,136)(61,104,105,83,79,98)(64,73,103,85,86,99)(68,119,120,91,92,94)(71,125,117,95,96,89)(82,141,142,109,110,177)(84,139,140,112,108,133)(87,102,138,114,115,134)(93,154,155,121,122,129)(97,161,159,126,127,124)(100,166,152,130,131,118)(111,182,183,144,145,224)(113,180,181,147,143,174)(116,137,179,149,150,175)(123,195,196,156,157,170)(128,202,203,162,163,165)(132,208,200,167,168,160)(135,213,193,171,172,153)(146,229,230,185,186,277)(148,227,228,188,184,221)(151,178,226,190,191,222)(158,242,243,197,198,217)(164,249,250,204,205,212)(169,256,254,209,210,207)(173,261,247,214,215,201)(176,266,240,218,219,194)(187,282,283,232,233,329)(189,280,281,235,231,274)(192,225,279,237,238,275)(199,293,294,244,245,270)(206,299,300,251,252,265)(211,304,305,257,258,260)(216,308,302,262,263,255)(220,313,297,267,268,248)(223,318,291,271,272,241)(234,306,312,259,285,345)(236,332,333,286,284,326)(239,278,331,288,289,327)(246,338,339,295,296,322)(253,334,342,287,301,317)(264,348,346,309,310,307)(269,350,343,314,315,303)(273,354,340,319,320,298)(276,358,336,323,324,292)(290,330,349,311,335,365)(316,373,370,351,352,347)(321,374,368,355,356,344)(325,375,366,359,360,341)(328,372,353,362,363,337)(357,379,380,367,364,371)(361,381,378,376,377,369);(6,7,8)(9,10,13,19,27,40,59,82,111,146,187,234,257,209,167,130,86,39,56,67,70,72,98,129,165,207,255,303,344,369,376,359,323,238,145,184,198,205,210,215,219,222,224,274,322,333,230,279,318,313,308,304,299,293,280,229,278,328,364,377,380,370,346,305,250,196,140,78,103,125,119,104,77,102,135,173,216,264,311,288,237,190,149,114,85,62,43,30,15,14,20,29,42,61,84,113,148,189,236,287,251,204,162,126,95,63,26,37,47,49,69,94,124,160,201,248,298,341,367,355,319,271,191,110,143,157,163,168,172,175,177,221,270,317,283,331,358,354,350,348,285,334,338,332,282,330,352,356,360,363,365,345,300,243,181,107,138,166,161,154,139,106,137,176,220,269,316,353,336,291,240,193,152,117,88,53,23,34,22,33,48,68,93,123,158,199,246,295,244,197,156,121,91,66,44,18,24,32,46,65,89,118,153,194,241,292,337,351,314,267,218,150,81,108,122,127,131,134,136,174,217,265,312,349,372,375,374,373,335,233,284,296,301,306,310,315,320,324,327,329,342,294,228,142,179,213,208,202,195,180,141,178,223,273,321,357,366,340,297,247,200,159,120,76,36,51,52,35,50,71,97,128,164,206,253,286,235,188,147,112,83,60,41,28,12,16,21,31,45,64,87,116,151,192,239,290,309,262,214,171,115,58,79,92,96,99,101,133,170,212,260,307,347,371,381,379,362,289,186,231,245,252,258,263,268,272,275,277,326,339,281,183,226,266,261,256,249,242,227,182,225,276,325,361,378,368,343,302,254,203,155,105,55,74,90,75,54,73,100,132,169,211,259,232,185,144,109,80,57,38,25,17,11)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33570, 17);

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