#27532837

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1794 ⋊ C₂
Sylow	C₅₃ ⋊ #32894

Subgroups

	subgroup	G/H
center	C₂	#20282993
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1794	C₂
socle	C₂×C₅₃	#2846
hypercenter	#1794	D₅₃
hypocenter	C₅₃	#32894
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1794

3638 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)(54,57)(55,63)(56,68)(58,74)(59,71)(60,72)(61,76)(62,64)(65,78)(66,79)(67,82)(69,84)(70,85)(73,83)(75,80)(77,81);(54,57)(55,62)(56,68)(58,71)(59,74)(60,72)(61,75)(63,64)(65,79)(66,78)(67,81)(69,83)(70,85)(73,84)(76,80)(77,82);(54,56)(55,61)(57,68)(58,73)(59,69)(60,70)(62,75)(63,76)(64,80)(65,67)(66,77)(71,84)(72,85)(74,83)(78,82)(79,81);(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)(56,67)(57,63)(58,65)(59,64)(60,66)(61,69)(62,71)(68,75)(70,80)(72,79)(73,77)(74,78)(76,83)(81,84)(82,85)

Polycyclic presentation:

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

Conjugacy classes & character table

878 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 318313);