#27197464

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

Aliases

No aliases for this group.

Splittings

Fitting	M₆₄×C₁₀₃ ⋊ C₂
Sylow	C₁₀₃ ⋊ #2264

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₆₄₈
commutator, cosocle	C₁₆×C₁₀₃	C₂³
Frattini	C₁₆	C₂²×D₁₀₃
Fitting	M₆₄×C₁₀₃	C₂
socle	C₂×C₁₀₃	C₂×D₁₆
prime radical	#20268223	C₂
hypercenter	M₆₄	D₁₀₃
hypocenter	C₁₀₃	#2264
LCS	G ▷ C₁₆×C₁₀₃ ▷ C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

104 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:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(105,112)(111,133)(113,136)(114,137)(115,131)(116,143)(121,142)(127,161)(134,163)(135,141)(138,164)(139,152)(140,165)(144,151)(148,167)(153,159);(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)(104,106,110,120)(105,111,112,133)(107,122,126,156)(108,128,130,162)(109,132,129,119)(113,138,136,164)(114,140,137,165)(115,141,131,135)(116,142,143,121)(117,145,147,124)(118,149,150,123)(125,160,158,155)(127,134,161,163)(139,167,152,148)(144,153,151,159)(146,157,154,166);(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)(104,105)(106,116)(107,121)(108,127)(109,131)(110,112)(111,117)(113,124)(114,123)(115,129)(118,148)(119,151)(120,143)(122,140)(125,159)(126,142)(128,138)(130,161)(132,144)(133,147)(134,155)(135,154)(136,145)(137,149)(139,166)(141,146)(150,167)(152,157)(153,158)(156,165)(160,163)(162,164)

Polycyclic presentation:

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

Conjugacy classes & character table

2066 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 6830);