#27369277

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1683 ⋊ C₂
Sylow	C₅₃ ⋊ #19387

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1683	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂×D₄×D₅₃	D₄
hypercenter	#1683	D₅₃
hypocenter	C₅₃	#19387
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1683

654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

2^8,53|a^2=g,b^2=eg,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dh,i^b=i^52,f^c=fh,i^d=i^52

Conjugacy classes & character table

803 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 154753);