#27209929

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	#578×C₁₀₃ ⋊ C₂
Sylow	C₁₀₃ ⋊ #3586

Subgroups

	subgroup	G/H
center	C₂	C₂²×D₄×D₁₀₃
commutator, cosocle	C₄×C₁₀₃	C₂⁵
Frattini	C₄	C₂⁴×D₁₀₃
Fitting	#578×C₁₀₃	C₂
socle	C₂×C₁₀₃	C₂³×D₄
hypercenter	#578	D₁₀₃
hypocenter	C₁₀₃	#3586
LCS	G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #578

1412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 135
transitive degree	μ_t(G) = 3296

A minimal faithful permutation representation — 5 generators on 135 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)(105,112)(110,124)(111,125)(113,120)(114,127)(123,134)(126,132)(128,135);(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)(107,118)(111,125)(115,129)(119,131)(123,134)(126,132)(128,135)(130,133);(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)(104,107)(105,111)(106,115)(108,119)(109,118)(110,123)(112,125)(113,126)(114,128)(116,130)(117,129)(120,132)(121,133)(122,131)(124,134)(127,135);(104,106,109,117)(105,110,112,124)(107,115,118,129)(108,121,122,116)(111,123,125,134)(113,127,120,114)(119,133,131,130)(126,135,132,128);(104,105)(106,114)(107,111)(108,120)(109,112)(110,116)(113,122)(115,128)(117,127)(118,125)(119,132)(121,124)(123,130)(126,131)(129,135)(133,134)

Polycyclic presentation:

2^7,103|b^2=g,f^2=g,b^a=bf,e^a=eg,f^a=fg,f^b=fg,d^c=dg,h^c=h^102,h^d=h^102,h^e=h^102

Conjugacy classes & character table

1163 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 19295);