#34095247

order	20032
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₃₁₃ ⋊ #346

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂×#14578818
commutator, socle	C₂×C₃₁₃	C₄×C₈
Fitting	C₈×C₃₁₃	C₈
prime radical	C₂×D₃₁₃	C₂×C₈
cosocle	#19097569	C₂²
hypercenter	C₈	#14578818
hypocenter	C₃₁₃	#346
LCS	G ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₄ ◁ C₈

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 329 points:

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

Polycyclic presentation:

2^6,313|a^2=c,b^2=d,c^2=e,e^2=f,b^a=bf,g^a=g^5,g^b=g^288,g^c=g^25,g^d=g^312,g^e=g^312

Conjugacy classes & character table

352 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20032, 1644);