#27197454

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	Q₆₄×C₁₀₃ ⋊ C₂
Sylow	C₁₀₃ ⋊ #2262

Subgroups

	subgroup	G/H
center	C₂	D₁₆×D₁₀₃
commutator, cosocle	C₁₆×C₁₀₃	C₂³
Frattini	C₁₆	C₂²×D₁₀₃
Fitting	Q₆₄×C₁₀₃	C₂
socle	C₂×C₁₀₃	C₂×D₁₆
hypercenter	Q₆₄	D₁₀₃
hypocenter	C₁₀₃	#2262
LCS	G ▷ C₁₆×C₁₀₃ ▷ C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 167
transitive degree	μ_t(G) = 6592

A minimal faithful permutation representation — 3 generators on 167 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)(104,107,110,125)(105,112,116,115)(106,118,122,121)(108,123,129,156)(109,124,132,157)(111,133,135,143)(113,136,140,139)(114,137,142,130)(117,144,145,152)(119,146,149,148)(120,147,151,131)(126,153,159,158)(127,154,161,160)(128,155,162,165)(134,141,163,167)(138,150,164,166);(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)(104,106)(105,111)(107,118)(108,127)(109,131)(110,122)(112,133)(113,138)(114,141)(115,143)(116,135)(117,126)(119,128)(120,124)(121,125)(123,154)(129,161)(130,134)(132,147)(136,150)(137,163)(139,166)(140,164)(142,167)(144,153)(145,159)(146,155)(148,165)(149,162)(151,157)(152,158)(156,160);(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)(106,117)(107,112)(108,126)(109,130)(110,116)(111,119)(113,128)(114,124)(115,125)(118,144)(120,150)(121,152)(122,145)(123,153)(127,141)(129,159)(131,138)(132,137)(133,146)(134,160)(135,149)(136,155)(139,165)(140,162)(142,157)(143,148)(147,164)(151,166)(154,163)(156,158)(161,167)

Polycyclic presentation:

2^7,103|c^2=g,d^2=ef,e^2=fg,f^2=g,b^a=bd,d^a=de,e^a=ef,f^a=fg,h^a=h^102,d^b=de,e^b=ef,f^b=fg,h^b=h^102,h^c=h^102

Conjugacy classes & character table

1007 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 6820);