C₂×D₃×C₃₆₁⋊C₃ #27188692

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

Aliases

direct product	C₂×D₃×C₃₆₁⋊C₃
unambiguous semidirect product	C₃⋊(C₂²×C₃₆₁⋊C₃)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₃₆₁⋊C₃)
unambiguous semidirect product	(C₂×D₃×C₃₆₁)⋊C₃

Splittings

Fitting	C₂×C₃×C₃₆₁ ⋊ C₂×C₃
derived	C₃×C₃₆₁ ⋊ C₂²×C₃
Hall	C₃×C₃₆₁⋊C₃ ⋊ C₂²
Sylow	C₃₆₁ ⋊ C₂×C₃×D₃

Subgroups

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

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 366
transitive degree	μ_t(G) = 2166

A minimal faithful permutation representation — 2 generators on 366 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12996, 35);

C2×D3×C361⋊C3 #27188692