C₃×Dic₁₀₈₃ #27188682

order	12996
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×Dic₁₀₈₃
unambiguous semidirect product	C₃⋊(C₃×Dic₃₆₁)

Splittings

derived	C₃×C₃₆₁ ⋊ C₃×C₄
Hall	C₃²×C₃₆₁ ⋊ C₄
Sylow	C₃₆₁ ⋊ C₃×Dic₃
Sylow	C₃² ⋊ Dic₃₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₁₀₈₃
commutator, hypocenter	C₃×C₃₆₁	C₃×C₄
Frattini	C₂×C₁₉	C₃×D₅₇
Fitting	C₂×C₃²×C₃₆₁	C₂
socle, prime radical	C₂×C₃²×C₁₉	D₁₉
cosocle	C₂×C₃×C₃₆₁	C₂×C₃
LCS	G ▷ C₃×C₃₆₁
UCS	C₁ ◁ C₂×C₃

42 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 371
transitive degree	μ_t(G) = 4332

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

Conjugacy classes & character table

3258 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12996, 25);

C3×Dic1083 #27188682