#27411914

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2047 ⋊ C₂
Sylow	C₅₃ ⋊ #23522

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2047	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂²×#11719949	C₂²
hypercenter	#2047	D₅₃
hypocenter	C₅₃	#23522
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2047

1326 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)(59,66)(64,80)(65,72)(67,76)(68,83)(79,81)(82,87)(84,85);(55,57)(58,61,62,74,63,75,78,89)(59,65,67,82,66,72,76,87)(60,69,70,86,71,73,77,88)(64,79,68,84,80,81,83,85);(58,60,63,71)(59,64,66,80)(61,73,75,69)(62,77,78,70)(65,81,72,79)(67,83,76,68)(74,86,89,88)(82,84,87,85);(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)(54,55)(56,57)(58,59)(60,68)(61,72)(62,76)(63,66)(64,70)(65,75)(67,78)(69,85)(71,83)(73,84)(74,82)(77,80)(79,88)(81,86)(87,89)

Polycyclic presentation:

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

Conjugacy classes & character table

722 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 197390);