#34095077

order	20032
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting	C₁₆×C₃₁₃ ⋊ C₄
Sylow	C₃₁₃ ⋊ #365

Subgroups

	subgroup	G/H
center	C₂	#23756565
commutator	C₈×C₃₁₃	C₂×C₄
Frattini	C₈	C₂×#11929411
Fitting	C₁₆×C₃₁₃	C₄
socle	C₂×C₃₁₃	#106
prime radical	C₂×D₃₁₃	SD₁₆
cosocle	C₈×D₃₁₃	C₂²
hypercenter	C₁₆	#11929411
hypocenter	C₃₁₃	#365
LCS	G ▷ C₈×C₃₁₃ ▷ C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

42 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:

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

Polycyclic presentation:

2^6,313|a^2=d,b^2=c,c^2=e,e^2=f,b^a=bc,c^a=ce,e^a=ef,g^a=g^288,d^b=df,g^d=g^312

Conjugacy classes & character table

1264 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20032, 1474);