#27387885

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1967 ⋊ C₂
Sylow	C₅₃ ⋊ #21220

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1967	C₂
socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
prime radical	C₂²×#11719947	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1967	D₅₃
hypocenter	C₅₃	#21220
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1967

1454 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)(73,82)(75,86)(79,88)(84,85)(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)(54,55,56,57)(58,61)(59,65)(60,69)(62,73)(63,72)(64,75)(66,78)(67,79)(68,82)(70,83)(71,84)(74,85)(76,87)(77,86)(80,88)(81,89);(58,60,68,71,63,70,62,74)(59,64,67,81,66,77,80,76)(61,69,82,84,72,83,73,85)(65,75,79,89,78,86,88,87);(55,57)(58,59,62,67,63,66,68,80)(60,64,71,76,70,77,74,81)(61,65,73,79,72,78,82,88)(69,75,84,87,83,86,85,89)

Polycyclic presentation:

2^8,53|a^2=g,b^2=egh,c^2=f,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=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, 173361);