#27182895

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	#472×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ #3288

Subgroups

	subgroup	G/H
center	C₂	#20265811
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#472×C₁₀₁	C₂
socle	C₂×C₁₀₁	D₄²
hypercenter	#472	D₁₀₁
hypocenter	C₁₀₁	#3288
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #472

464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 1616

A minimal faithful permutation representation — 4 generators on 117 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)(102,104)(103,109)(105,115)(106,112)(107,113)(108,116)(110,117)(111,114);(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)(104,109)(105,108)(106,114)(107,110)(111,112)(113,117)(115,116);(102,103)(104,111)(105,114)(106,108)(107,110)(109,112)(113,116)(115,117);(103,108)(105,106)(109,116)(110,114)(111,117)(112,115)

Polycyclic presentation:

2^7,101|e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=de,e^b=eg,f^c=fg,h^c=h^100,e^d=eg,h^d=h^100

Conjugacy classes & character table

826 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 16273);