C₃⋊D₃×D₃₆₁ #27188697

order	12996
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₃⋊D₃×D₃₆₁
unambiguous semidirect product	(C₃×D₃₆₁)⋊D₃

Splittings

Fitting, derived, Hall	C₃²×C₃₆₁ ⋊ C₂²
Sylow	C₃₆₁ ⋊ C₂×C₃⋊D₃
Sylow	C₃² ⋊ C₂×D₃₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₃²×C₃₆₁	C₂²
Frattini	C₁₉	C₃⋊D₃×D₁₉
socle	C₃²×C₁₉	C₂×D₁₉
LCS	G ▷ C₃²×C₃₆₁
UCS	C₁

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 367
transitive degree	μ_t(G) = 3249

A minimal faithful permutation representation — 3 generators on 367 points:

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

Conjugacy classes & character table

1092 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12996, 40);

C3⋊D3×D361 #27188697