#34102431

order	20096
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	M₆₄×C₁₅₇ ⋊ C₂
Sylow	C₁₅₇ ⋊ #2258

Subgroups

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₁₅₇
commutator, socle	C₂×C₁₅₇	C₂²×C₁₆
Fitting	M₆₄×C₁₅₇	C₂
prime radical	#14578954	C₈
cosocle	C₁₆×C₁₅₇	C₂³
hypercenter	M₆₄	D₁₅₇
hypocenter	C₁₅₇	#2258
LCS	G ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₁₆ ◁ M₆₄

106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 221
transitive degree	μ_t(G) = 10048

A minimal faithful permutation representation — 3 generators on 221 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)(160,170)(165,180)(171,190)(172,191)(173,192)(181,200)(182,201)(183,202)(193,210)(194,211)(195,212)(203,215)(204,216)(205,217)(213,220)(218,221);(158,160)(159,165)(161,171)(162,172)(163,173)(164,170)(166,181)(167,182)(168,183)(169,180)(174,193)(175,194)(176,190)(177,195)(178,191)(179,192)(184,203)(185,204)(186,200)(187,205)(188,201)(189,202)(196,213)(197,210)(198,211)(199,212)(206,218)(207,215)(208,216)(209,217)(214,220)(219,221);(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)(158,159,161,166,162,167,174,184,163,168,175,185,177,187,196,206,164,169,176,186,178,188,197,207,179,189,198,208,199,209,214,219)(160,165,171,181,172,182,193,203,173,183,194,204,195,205,213,218,170,180,190,200,191,201,210,215,192,202,211,216,212,217,220,221)

Polycyclic presentation:

2^7,157|a^2=d,d^2=e,e^2=f,f^2=g,h^a=h^156,c^b=cg,h^c=h^156

Conjugacy classes & character table

3200 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 6805);