#27472466

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1911 ⋊ C₂
Sylow	C₅₃ ⋊ #28182

Subgroups

	subgroup	G/H
center	C₂	#20279615
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1911	C₂
socle	C₂×C₅₃	#2535
prime radical	#20275768	C₂
hypercenter	#1911	D₅₃
hypocenter	C₅₃	#28182
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1911

788 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)(54,57,58,69,59,70,71,82)(55,62,60,73,63,77,74,84)(56,65,66,79,68,81,67,80)(61,75,72,83,76,85,64,78);(54,56)(55,61)(57,65)(58,66)(59,68)(60,72)(62,75)(63,76)(64,74)(67,71)(69,79)(70,81)(73,83)(77,85)(78,84)(80,82);(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)(54,55,58,60,59,63,71,74)(56,64,67,76,68,72,66,61)(57,62,69,73,70,77,82,84)(65,78,80,85,81,83,79,75);(55,60)(57,69)(58,71)(61,64)(62,77)(63,74)(65,79)(66,67)(70,82)(72,76)(78,83)(80,81)

Polycyclic presentation:

2^8,53|b^2=g,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,i^b=i^52,g^c=gh,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, 257942);