#27402579

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1860 ⋊ C₂
Sylow	C₅₃ ⋊ #22691

Subgroups

	subgroup	G/H
center	C₂²	#16811255
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1860	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#13133237	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1860	D₅₃
hypocenter	C₅₃	#22691
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1860

846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 188055);