#27213564

order	13504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	M₃₂×C₂₁₁ ⋊ C₂
Sylow	C₂₁₁ ⋊ #504

Subgroups

	subgroup	G/H
center, Frattini	C₈	C₂²×D₂₁₁
commutator, socle	C₂×C₂₁₁	C₂²×C₈
Fitting	M₃₂×C₂₁₁	C₂
prime radical	#16809962	C₄
cosocle	C₈×C₂₁₁	C₂³
hypercenter	M₃₂	D₂₁₁
hypocenter	C₂₁₁	#504
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₈ ◁ M₃₂

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 243
transitive degree	μ_t(G) = 6752

A minimal faithful permutation representation — 3 generators on 243 points:

(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90)(91,92)(93,94)(95,96)(97,98)(99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)(209,210)(214,222)(218,228)(223,234)(224,235)(229,238)(230,239)(236,242)(240,243);(212,214)(213,218)(215,223)(216,224)(217,222)(219,229)(220,230)(221,228)(225,236)(226,234)(227,235)(231,240)(232,238)(233,239)(237,242)(241,243);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(98,99)(100,101)(102,103)(104,105)(106,107)(108,109)(110,111)(112,113)(114,115)(116,117)(118,119)(120,121)(122,123)(124,125)(126,127)(128,129)(130,131)(132,133)(134,135)(136,137)(138,139)(140,141)(142,143)(144,145)(146,147)(148,149)(150,151)(152,153)(154,155)(156,157)(158,159)(160,161)(162,163)(164,165)(166,167)(168,169)(170,171)(172,173)(174,175)(176,177)(178,179)(180,181)(182,183)(184,185)(186,187)(188,189)(190,191)(192,193)(194,195)(196,197)(198,199)(200,201)(202,203)(204,205)(206,207)(208,209)(210,211)(212,213,215,219,216,220,225,231,217,221,226,232,227,233,237,241)(214,218,223,229,224,230,236,240,222,228,234,238,235,239,242,243)

Polycyclic presentation:

2^6,211|a^2=d,d^2=e,e^2=f,g^a=g^210,c^b=cf,g^c=g^210

Conjugacy classes & character table

2140 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13504, 967);