#27444589

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2078 ⋊ C₂
Sylow	C₅₃ ⋊ #26193

Subgroups

	subgroup	G/H
center	C₂²	#16811257
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2078	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#13133237	C₂²
hypercenter	#2078	D₅₃
hypocenter	C₅₃	#26193
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2078

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(62,66,68,65)(63,71,74,90)(64,77,79,80)(67,81,85,82)(69,86,88,93)(70,73,83,72)(75,91,87,89)(76,84,92,78);(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)(54,57,58,56)(55,59,60,61)(62,65,68,66)(63,70,74,83)(64,76,79,92)(67,81,85,82)(69,86,88,93)(71,73,90,72)(75,89,87,91)(77,84,80,78);(54,56,58,57)(55,59,60,61)(62,64)(63,69)(65,80)(66,77)(67,84)(68,79)(70,89)(71,86)(72,87)(73,75)(74,88)(76,82)(78,85)(81,92)(83,91)(90,93);(54,55)(56,59)(57,61)(58,60)(62,63)(64,75)(65,70)(66,83)(67,73)(68,74)(69,78)(71,82)(72,85)(76,89)(77,86)(79,87)(80,93)(81,90)(84,88)(91,92)

Polycyclic presentation:

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

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 230065);