C₂×D₃×C₃₆₇⋊C₃ #27210209

order	13212
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×D₃×C₃₆₇⋊C₃
unambiguous semidirect product	C₃⋊(C₂²×C₃₆₇⋊C₃)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₃₆₇⋊C₃)
unambiguous semidirect product	(C₂×D₃×C₃₆₇)⋊C₃

Splittings

Fitting, socle	C₂×C₃×C₃₆₇ ⋊ C₂×C₃
derived	C₃×C₃₆₇ ⋊ C₂²×C₃
Hall	C₃×C₃₆₇⋊C₃ ⋊ C₂²
Sylow	C₃₆₇ ⋊ C₂×C₃×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃×C₃₆₇⋊C₃
commutator, cosocle, hypocenter	C₃×C₃₆₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₃₆₇	C₂×C₃
LCS	G ▷ C₃×C₃₆₇
UCS	C₁ ◁ C₂

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 372
transitive degree	μ_t(G) = 2202

A minimal faithful permutation representation — 2 generators on 372 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13212, 31);

C2×D3×C367⋊C3 #27210209