#27421474

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3041 ⋊ C₂
Sylow	C₅₃ ⋊ #24408

Subgroups

	subgroup	G/H
center	C₂	#20278086
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3041	C₂
socle	C₂×C₅₃	#2410
hypercenter	#3041	D₅₃
hypocenter	C₅₃	#24408
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3041

1364 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)(55,60)(57,65)(58,67)(59,66)(61,74)(62,64)(63,75)(68,70)(69,83)(71,73)(72,84)(76,77)(78,82)(79,80)(81,85);(54,57)(55,61)(56,65)(58,72)(59,69)(60,74)(62,78)(63,81)(64,82)(66,83)(67,84)(68,71)(70,73)(75,85)(76,80)(77,79);(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)(55,60)(57,68)(58,71)(59,66)(61,76)(62,64)(63,80)(65,70)(67,73)(69,84)(72,83)(74,77)(75,79)(78,85)(81,82);(54,55)(56,64,66,60)(57,61)(58,63)(59,62)(65,74,83,82)(67,80,71,75)(68,76,84,85)(69,78)(70,77)(72,81)(73,79)

Polycyclic presentation:

2^8,53|a^2=f,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gh,i^b=i^52,e^c=eh,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, 206950);