#27415416

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3028 ⋊ C₂
Sylow	C₅₃ ⋊ #23875

Subgroups

	subgroup	G/H
center	C₂	#20278026
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3028	C₂
socle	C₂×C₅₃	#2403
hypercenter	#3028	D₅₃
hypocenter	C₅₃	#23875
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3028

1642 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)(54,58)(55,63)(56,67)(57,69)(59,75)(60,73)(61,77)(62,79)(64,81)(65,80)(66,82)(68,84)(70,83)(71,74)(72,76)(78,85);(54,57,60,56)(55,62,65,66)(58,69,73,67)(59,68,76,70)(61,64,78,74)(63,79,80,82)(71,77,81,85)(72,83,75,84);(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)(54,56,60,57)(55,61,65,78)(58,67,73,69)(59,70,76,68)(62,74,66,64)(63,77,80,85)(71,82,81,79)(72,84,75,83);(54,55)(56,66)(57,62)(58,71)(59,74)(60,65)(61,68)(63,72)(64,76)(67,77)(69,85)(70,78)(73,81)(75,80)(79,84)(82,83)

Polycyclic presentation:

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

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 200892);