#27197439

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	M₆₄×C₁₀₃ ⋊ C₂
Sylow	C₁₀₃ ⋊ #2258

Subgroups

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₁₀₃
commutator, socle	C₂×C₁₀₃	C₂²×C₁₆
Fitting	M₆₄×C₁₀₃	C₂
prime radical	#13110105	C₈
cosocle	C₁₆×C₁₀₃	C₂³
hypercenter	M₆₄	D₁₀₃
hypocenter	C₁₀₃	#2258
LCS	G ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₁₆ ◁ M₆₄

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

Polycyclic presentation:

2^7,103|a^2=d,d^2=e,e^2=f,f^2=g,h^a=h^102,c^b=cg,h^c=h^102

Conjugacy classes & character table

2120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 6805);