#34098321

order	20096
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃₂×C₁₅₇ ⋊ C₂
Sylow	C₁₅₇ ⋊ #1417

Subgroups

	subgroup	G/H
center	C₄	D₂₅₁₂
commutator	C₁₆×C₁₅₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₅₇
Fitting	C₂×C₃₂×C₁₅₇	C₂
socle	C₂×C₁₅₇	#357
prime radical	#23757705	C₂
cosocle	C₂×C₁₆×C₁₅₇	C₂²
hypercenter	C₂×C₃₂	D₁₅₇
hypocenter	C₁₅₇	#1417
LCS	G ▷ C₁₆×C₁₅₇ ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 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)(158,160)(159,165)(161,178)(162,173)(163,185)(164,176)(166,192)(167,188)(168,197)(169,199)(170,190)(171,177)(172,181)(174,179)(175,183)(180,211)(182,213)(184,189)(186,204)(187,191)(193,205)(194,220)(195,219)(196,217)(198,221)(200,208)(201,210)(202,218)(203,216)(206,214)(207,209)(212,215);(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)(158,159,162,167,164,170,183,169)(160,171,173,200,176,201,175,207)(161,177,180,208,182,210,215,209)(163,184,179,168,186,196,214,198)(165,187,188,203,190,202,199,172)(166,191,193,216,195,218,194,181)(174,205,204,219,206,220,185,192)(178,197,211,217,213,221,212,189)

Polycyclic presentation:

2^7,157|a^2=d,c^2=ef,d^2=g,e^2=fg,f^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,h^a=h^156,c^b=ce,e^b=ef,f^b=fg,h^b=h^156

Conjugacy classes & character table

5030 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 2695);