#27359622

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1614 ⋊ C₂
Sylow	C₅₃ ⋊ #18467

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1614	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20289282	C₂
hypercenter	#1614	D₅₃
hypocenter	C₅₃	#18467
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1614

890 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)(54,57,59,58)(55,61,62,66)(56,64,65,67)(60,68,69,63)(70,72)(71,76)(73,79)(74,78)(75,82)(77,84)(80,85)(81,83);(57,58)(61,66)(63,68)(64,67)(72,78)(76,84)(79,85)(82,83);(54,56,59,65)(55,60,62,69)(57,64,58,67)(61,68,66,63)(70,71,74,77)(72,76,78,84)(73,81,80,75)(79,83,85,82);(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)(54,55,59,62)(56,63,65,68)(57,61,58,66)(60,67,69,64)(71,75)(72,78)(73,80)(76,83)(77,81)(82,84)

Polycyclic presentation:

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

Conjugacy classes & character table

632 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 145098);