#27590790

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2820 ⋊ C₂
Sylow	C₅₃ ⋊ #36699

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator, socle, cosocle	C₂³×C₅₃	C₂⁵
Fitting	C₅₃×#2820	C₂
prime radical	C₂×#13133188	C₂²
hypercenter	#2820	D₅₃
hypocenter	C₅₃	#36699
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2820

1958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

2^8,53|a^2=gh,b^2=h,e^2=f,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, 376266);