D₉×D₃₆₁ #27188666

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

Aliases

direct product	D₉×D₃₆₁
unambiguous semidirect product	(D₉×C₃₆₁)⋊C₂
unambiguous semidirect product	(C₉×D₃₆₁)⋊C₂
unambiguous semidirect product	D₃₂₄₉⋊C₂

Splittings

Fitting, derived, Hall	C₉×C₃₆₁ ⋊ C₂²
Sylow	C₃₆₁ ⋊ C₂×D₉
Sylow	C₉ ⋊ C₂×D₃₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₃₆₁	C₂²
Frattini, socle	C₃×C₁₉	D₃×D₁₉
LCS	G ▷ C₉×C₃₆₁
UCS	C₁

45 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 370
transitive degree	μ_t(G) = 3249

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

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, 9);

D9×D361 #27188666