#27550264

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #34530

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Fitting	C₅₃×#2330	C₂
socle, prime radical, cosocle	C₂³×C₅₃	C₂⁵
hypercenter	#2330	D₅₃
hypocenter	C₅₃	#34530
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #2330

1800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 13568

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

Polycyclic presentation:

2^8,53|a^2=h,b^2=g,c^2=fh,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,i^b=i^52,i^d=i^52,i^e=i^52

Conjugacy classes & character table

1220 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 335740);