#34095274

order	20032
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃₁₃ ⋊ #504

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×#14578818
commutator, socle	C₂×C₃₁₃	C₂²×C₈
Fitting	Q₈×C₃₁₃	C₈
prime radical	#19097562	C₄
cosocle	#14578816	C₂³
hypercenter	Q₈	#14578818
hypocenter	C₃₁₃	#504
LCS	G ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂ ◁ Q₈

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 345
transitive degree	μ_t(G) = 10016

A minimal faithful permutation representation — 3 generators on 345 points:

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

Polycyclic presentation:

2^6,313|a^2=d,d^2=e,e^2=f,g^a=g^5,c^b=cf,g^b=g^312,g^c=g^312,g^d=g^25,g^e=g^312

Conjugacy classes & character table

235 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20032, 1671);