#27465204

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1938 ⋊ C₂
Sylow	C₅₃ ⋊ #27599

Subgroups

	subgroup	G/H
center	C₄	#16811189
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1938	C₂
socle	C₂×C₅₃	#2482
prime radical	#16811459	C₂²
hypercenter	#1938	D₅₃
hypocenter	C₅₃	#27599
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1938

720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 250680);