#27590815

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #36701

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, prime radical, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2839	C₂
hypercenter	#2839	D₅₃
hypocenter	C₅₃	#36701
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2839

1304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 13568

A minimal faithful permutation representation — 5 generators on 93 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)(54,56,57,60)(55,59,58,61)(78,82,83,81)(79,85,86,91)(80,88,90,89)(84,93,87,92);(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)(63,69)(65,73)(67,75)(70,72)(78,81,83,82)(79,85,86,91)(80,89,90,88)(84,93,87,92);(55,58)(59,61)(62,65)(63,68)(64,70)(66,73)(67,74)(69,77)(71,72)(75,76)(78,81,83,82)(79,85,86,91)(80,88,90,89)(84,92,87,93);(62,64,66,71)(63,67,69,75)(65,72,73,70)(68,76,77,74)(78,80,83,90)(79,84,86,87)(81,89,82,88)(85,93,91,92);(54,55)(56,59)(57,58)(60,61)(62,63,66,69)(64,67,71,75)(65,68,73,77)(70,74,72,76)(78,79,83,86)(80,87,90,84)(81,91,82,85)(88,92,89,93)

Polycyclic presentation:

2^8,53|a^2=h,b^2=h,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^c=dfh,i^d=i^52,i^e=i^52

Conjugacy classes & character table

728 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 376291);