#27420141

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3019 ⋊ C₂
Sylow	C₅₃ ⋊ #24297

Subgroups

	subgroup	G/H
center	C₂	#20278068
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3019	C₂
socle	C₂×C₅₃	#2408
hypercenter	#3019	D₅₃
hypocenter	C₅₃	#24297
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3019

1334 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)(54,58,60,72,61,73,59,71)(55,63,64,78,65,79,56,67)(57,68,74,84,70,83,69,82)(62,75,76,85,77,81,66,80);(54,57)(55,62)(56,66)(58,68)(59,69)(60,74)(61,70)(63,75)(64,76)(65,77)(67,80)(71,82)(72,84)(73,83)(78,85)(79,81);(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)(54,56,59,65,61,64,60,55)(57,62,69,66,70,77,74,76)(58,67,71,79,73,78,72,63)(68,75,82,80,83,81,84,85);(54,55,61,65)(56,59,64,60)(57,66,70,76)(58,67,73,78)(62,74,77,69)(63,72,79,71)(68,81,83,75)(80,84,85,82)

Polycyclic presentation:

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

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 205617);