#34095020

order	20032
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₈×C₃₁₃ ⋊ C₄
Sylow	C₃₁₃ ⋊ H(3,ℤ/4)

Subgroups

	subgroup	G/H
center	C₄	C₄×#11929411
commutator	C₄×C₃₁₃	C₄²
Frattini	C₂×C₄	C₂×#11929411
Fitting	C₂×C₈×C₃₁₃	C₄
socle	C₂×C₃₁₃	#95
prime radical	C₂²×D₃₁₃	D₄
cosocle	C₂×C₄×D₃₁₃	C₂²
hypercenter	C₂×C₈	#11929411
hypocenter	C₃₁₃	H(3,ℤ/4)
LCS	G ▷ C₄×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈

94 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:

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

Polycyclic presentation:

2^6,313|a^2=d,b^2=e,c^2=f,b^a=bc,e^a=ef,g^a=g^288,d^b=df,g^b=g^25,g^d=g^312,g^e=g^312

Conjugacy classes & character table

1270 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20032, 1417);