#27204970

order	13184
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Fitting	#476×C₁₀₃ ⋊ C₂
Sylow	C₁₀₃ ⋊ #3081

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₃
commutator, cosocle	C₂×C₄×C₁₀₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₃
Fitting	#476×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂²×D₄
prime radical	C₂×D₄×D₁₀₃	C₂²
hypercenter	#476	D₁₀₃
hypocenter	C₁₀₃	#3081
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 3296

A minimal faithful permutation representation — 4 generators on 127 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)(114,121)(117,125)(122,127)(124,126);(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)(113,118)(115,123)(117,124)(119,120)(122,127)(125,126);(104,106,107,111)(105,108,109,110)(112,114)(113,117)(115,122)(116,121)(118,124)(119,126)(120,125)(123,127);(104,105,107,109)(106,110,111,108)(112,113)(114,117)(115,118)(116,120)(119,123)(121,125)(122,124)(126,127)

Polycyclic presentation:

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

Conjugacy classes & character table

1001 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13184, 14336);