C₃×D₆₀₀₁ #49112780

order	36006
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₆₀₀₁
unambiguous semidirect product	C₁₇⋊(C₃×D₃₅₃)
unambiguous semidirect product	C₃₅₃⋊(C₃×D₁₇)

Splittings

Fitting, socle, Hall	C₃×C₁₇×C₃₅₃ ⋊ C₂
derived, Hall	C₁₇×C₃₅₃ ⋊ C₂×C₃
Hall	C₃×C₃₅₃ ⋊ D₁₇
Sylow	C₃₅₃ ⋊ C₃×D₁₇
Hall	C₃×C₁₇ ⋊ D₃₅₃
Sylow	C₁₇ ⋊ C₃×D₃₅₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₆₀₀₁
commutator, cosocle, hypocenter	C₁₇×C₃₅₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₇×C₃₅₃	C₂
LCS	G ▷ C₁₇×C₃₅₃
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 373
transitive degree	μ_t(G) = 18003

A minimal faithful permutation representation — 2 generators on 373 points:

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

Conjugacy classes & character table

9006 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36006, 4);

C3×D6001 #49112780