#27383265

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1785 ⋊ C₂
Sylow	C₅₃ ⋊ #20593

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1785	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂²×#13133264	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1785	D₅₃
hypocenter	C₅₃	#20593
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1785

3594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 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)(61,72)(65,78)(69,83)(73,85)(76,82)(79,88)(84,86)(87,89);(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)(55,57)(58,61)(59,65)(60,69)(62,73)(63,72)(64,76)(66,78)(67,79)(68,82)(70,84)(71,83)(74,86)(75,85)(77,87)(80,88)(81,89);(58,60,63,71)(59,64,66,68)(61,69,72,83)(62,74,75,70)(65,76,78,82)(67,81,80,77)(73,86,85,84)(79,89,88,87);(54,55)(56,57)(58,59,62,67,63,66,75,80)(60,68,70,81,71,64,74,77)(61,65,73,79,72,78,85,88)(69,82,84,89,83,76,86,87)

Polycyclic presentation:

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

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 168741);