#27539356

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2230 ⋊ C₂
Sylow	C₅₃ ⋊ #33691

Subgroups

	subgroup	G/H
center	C₂	#20284808
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2230	C₂
socle	C₂×C₅₃	#3109
prime radical	#20276662	C₂
hypercenter	#2230	D₅₃
hypocenter	C₅₃	#33691
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2230

524 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)(55,61)(56,65,66,67)(57,71)(60,74,75,79)(63,76)(64,70,81,82)(68,84)(73,85,80,78);(54,56)(55,60)(57,70)(58,65)(59,66)(61,75)(62,78)(63,79)(64,68)(67,72)(69,85)(71,82)(73,77)(74,76)(80,83)(81,84);(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)(56,60)(57,69)(58,63)(59,61)(62,71)(64,80)(65,79)(66,75)(67,74)(68,83)(70,85)(72,76)(73,81)(77,84)(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)(56,64)(57,68)(58,72)(60,73)(62,77)(63,76)(65,70)(66,81)(67,82)(69,83)(71,84)(74,85)(75,80)(78,79)

Polycyclic presentation:

2^8,53|d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,i^a=i^52,d^b=dh,f^b=fh,i^b=i^52,d^c=dg,e^c=eh,f^c=fg,g^c=gh,f^d=fh,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, 324832);