C₅×C₃⋊D₁₁₁₉ #46777080

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

Aliases

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

Splittings

Fitting, socle, Hall	C₃²×C₅×C₃₇₃ ⋊ C₂
derived, Hall	C₃²×C₃₇₃ ⋊ C₂×C₅
Hall	C₅×C₃₇₃ ⋊ C₃⋊D₃
Sylow	C₃₇₃ ⋊ C₅×C₃⋊D₃
Hall	C₃²×C₅ ⋊ D₃₇₃
Sylow	C₃² ⋊ C₅×D₃₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊D₁₁₁₉
commutator, cosocle, hypocenter	C₃²×C₃₇₃	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₃₇₃	C₂
LCS	G ▷ C₃²×C₃₇₃
UCS	C₁ ◁ C₅

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 384
transitive degree	μ_t(G) = 16785

A minimal faithful permutation representation — 3 generators on 384 points:

(1,2,3,4,5)(7,8)(10,11)(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)(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)(335,336)(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)(351,352)(353,354)(355,356)(357,358)(359,360)(361,362)(363,364)(365,366)(367,368)(369,370)(371,372)(373,374)(375,376)(377,378)(379,380)(381,382)(383,384);(9,10,11)(12,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127,129,131,133,135,137,139,141,143,145,147,149,151,153,155,157,159,161,163,165,167,169,171,173,175,177,179,181,183,185,187,189,191,193,195,197,199,201,203,205,207,209,211,213,215,217,219,221,223,225,227,229,231,233,235,237,239,241,243,245,247,249,251,253,255,257,259,261,263,265,267,269,271,273,275,277,279,281,283,285,287,289,291,293,295,297,299,301,303,305,307,309,311,313,315,317,319,321,323,325,327,329,331,333,335,337,339,341,343,345,347,349,351,353,355,357,359,361,363,365,367,369,371,373,375,377,379,381,383,384,382,380,378,376,374,372,370,368,366,364,362,360,358,356,354,352,350,348,346,344,342,340,338,336,334,332,330,328,326,324,322,320,318,316,314,312,310,308,306,304,302,300,298,296,294,292,290,288,286,284,282,280,278,276,274,272,270,268,266,264,262,260,258,256,254,252,250,248,246,244,242,240,238,236,234,232,230,228,226,224,222,220,218,216,214,212,210,208,206,204,202,200,198,196,194,192,190,188,186,184,182,180,178,176,174,172,170,168,166,164,162,160,158,156,154,152,150,148,146,144,142,140,138,136,134,132,130,128,126,124,122,120,118,116,114,112,110,108,106,104,102,100,98,96,94,92,90,88,86,84,82,80,78,76,74,72,70,68,66,64,62,60,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24,22,20,18,16,14);(6,7,8)

Conjugacy classes & character table

8400 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33570, 30);

C5×C3⋊D1119 #46777080