C₃×C₃⋊#13093637 #27212009

order	13428
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₃⋊#13093637
unambiguous semidirect product	C₃⋊(C₃×#13093637)

Splittings

Fitting, socle, Hall	C₃²×C₃₇₃ ⋊ C₄
derived	C₃×C₃₇₃ ⋊ C₃×C₄
Sylow	C₃₇₃ ⋊ C₃×Dic₃
Sylow	C₃² ⋊ #13093637

Subgroups

	subgroup	G/H
center, hypercenter	C₃	C₃⋊#13093637
commutator, hypocenter	C₃×C₃₇₃	C₃×C₄
Frattini	C₁	G
Fitting, socle	C₃²×C₃₇₃	C₄
prime radical	C₃²×D₃₇₃	C₂
cosocle	C₃×D₃₇₃	C₂×C₃
LCS	G ▷ C₃×C₃₇₃
UCS	C₁ ◁ C₃

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 379
transitive degree	μ_t(G) = 2238

A minimal faithful permutation representation — 2 generators on 379 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13428, 38);

C3×C3⋊#13093637 #27212009