C₃×D₄₀₃₇ #37564233

order	24222
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₄₀₃₇
unambiguous semidirect product	C₁₁⋊(C₃×D₃₆₇)

Splittings

Fitting, socle, Hall	C₃×C₁₁×C₃₆₇ ⋊ C₂
derived, Hall	C₁₁×C₃₆₇ ⋊ C₂×C₃
Hall	C₃×C₃₆₇ ⋊ D₁₁
Sylow	C₃₆₇ ⋊ C₃×D₁₁
Hall	C₃×C₁₁ ⋊ D₃₆₇
Sylow	C₁₁ ⋊ C₃×D₃₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₄₀₃₇
commutator, cosocle, hypocenter	C₁₁×C₃₆₇	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₁×C₃₆₇	C₂
LCS	G ▷ C₁₁×C₃₆₇
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 381
transitive degree	μ_t(G) = 12111

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

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

Conjugacy classes & character table

6060 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24222, 5);

C3×D4037 #37564233