#27169290

order	12928
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Fitting	#355×C₁₀₁ ⋊ C₂
Sylow	C₁₀₁ ⋊ #1406

Subgroups

	subgroup	G/H
center	C₂	#20264969
commutator	C₄²×C₁₀₁	C₂×C₄
Frattini	#124	C₂×D₁₀₁
Fitting	#355×C₁₀₁	C₂
socle	C₂×C₁₀₁	#353
prime radical	#20265567	C₂
cosocle	#124×C₁₀₁	C₂²
hypercenter	#355	D₁₀₁
hypocenter	C₁₀₁	#1406
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #355

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 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)(103,107)(104,111)(105,114)(108,110)(112,113)(115,116);(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)(105,108,111,116)(106,110,112,117)(107,113,115,114)

Polycyclic presentation:

2^7,101|a^2=d,c^2=f,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,h^a=h^100,c^b=cf,d^b=defg,e^b=eg,h^b=h^100,d^c=dfg,e^d=eg

Conjugacy classes & character table

664 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 2668);