#27411316

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2020 ⋊ C₂
Sylow	C₅₃ ⋊ #23440

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2020	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂×#16811422	C₂
hypercenter	#2020	D₅₃
hypocenter	C₅₃	#23440
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2020

1898 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)(59,67)(65,81)(66,84)(68,69)(70,82)(75,85)(80,86)(83,89);(58,61,64,79)(59,66,67,84)(60,71,74,88)(62,78,63,77)(65,80,81,86)(68,85,69,75)(70,83,82,89)(72,76,73,87);(55,57)(58,60)(59,65)(61,76)(62,73)(63,72)(64,74)(66,83)(67,81)(68,82)(69,70)(71,78)(75,86)(77,88)(79,87)(80,85)(84,89);(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)(60,70)(61,75)(62,69)(63,68)(64,67)(65,72)(66,77)(71,86)(73,81)(74,82)(76,83)(78,84)(79,85)(80,88)(87,89)

Polycyclic presentation:

2^8,53|c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=gh,i^a=i^52,c^b=cg,e^b=eh,g^b=gh,e^c=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, 196792);