#34095194

order	20032
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	D₈×C₃₁₃ ⋊ C₄
Sylow	C₃₁₃ ⋊ #443

Subgroups

	subgroup	G/H
center	C₂	D₄×#11929411
commutator	C₄×C₃₁₃	C₂²×C₄
Frattini	C₄	C₂²×#11929411
Fitting	D₈×C₃₁₃	C₄
socle	C₂×C₃₁₃	C₄×D₄
prime radical	#23756516	C₂
cosocle	C₄×D₃₁₃	C₂³
hypercenter	D₈	#11929411
hypocenter	C₃₁₃	#443
LCS	G ▷ C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 329
transitive degree	μ_t(G) = 5008

A minimal faithful permutation representation — 3 generators on 329 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

574 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20032, 1591);