#27365970

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1664 ⋊ C₂
Sylow	C₅₃ ⋊ #19060

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1664	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20289166	C₂
hypercenter	#1664	D₅₃
hypocenter	C₅₃	#19060
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1664

822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

554 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 151446);