#27359487

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1605 ⋊ C₂
Sylow	C₅₃ ⋊ #18455

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1605	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#16811136	C₂²
hypercenter	#1605	D₅₃
hypocenter	C₅₃	#18455
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1605

894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 101 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)(55,61)(59,66)(60,67)(65,68)(70,74,76,75)(71,79,81,80)(72,84,85,89)(73,86,87,88)(77,82,91,95)(78,92,93,94)(83,97,99,98)(90,100,101,96);(54,57)(55,60)(56,62)(58,64)(59,65)(61,67)(63,69)(66,68)(70,73,76,87)(71,78,81,93)(72,83,85,99)(74,88,75,86)(77,90,91,101)(79,94,80,92)(82,96,95,100)(84,98,89,97);(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)(54,56)(55,59)(57,62)(58,63)(60,65)(61,66)(64,69)(67,68)(70,72)(71,77)(73,83)(74,84)(75,89)(76,85)(78,90)(79,82)(80,95)(81,91)(86,97)(87,99)(88,98)(92,100)(93,101)(94,96);(54,55)(56,59)(57,60)(58,61)(62,68)(63,66)(64,67)(65,69)(70,71,76,81)(72,82,85,95)(73,78,87,93)(74,80,75,79)(77,89,91,84)(83,96,99,100)(86,94,88,92)(90,97,101,98)

Polycyclic presentation:

2^8,53|a^2=g,c^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,i^b=i^52,d^c=dg,e^c=eh,i^d=i^52

Conjugacy classes & character table

632 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 144963);