#27422164

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2051 ⋊ C₂
Sylow	C₅₃ ⋊ #24458

Subgroups

	subgroup	G/H
center	C₂²	#16811257
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2051	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#16811433	C₂
hypercenter	#2051	D₅₃
hypocenter	C₅₃	#24458
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2051

1328 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)(58,61)(59,66)(60,70)(62,75)(63,76)(64,73)(65,79)(67,83)(68,82)(69,84)(71,87)(72,86)(74,77)(78,85)(80,88)(81,89);(55,57)(58,60,63,72)(59,65,68,81)(61,70,76,86)(62,71,77,69)(64,78,67,80)(66,79,82,89)(73,85,83,88)(74,84,75,87);(59,64)(60,69)(62,77)(66,73)(67,68)(70,84)(71,72)(74,75)(78,80)(82,83)(85,88)(86,87);(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)(60,65)(61,73)(62,64)(63,68)(66,74)(67,77)(69,80)(70,85)(71,78)(72,81)(75,82)(76,83)(79,84)(86,88)(87,89)

Polycyclic presentation:

2^8,53|a^2=f,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,i^a=i^52,c^b=cg,e^b=eh,g^b=gh,g^c=gh,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, 207640);