#27359358

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅₃ ⋊ #18445

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1612	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₅₃×#538	C₂²
hypercenter	#1612	D₅₃
hypocenter	C₅₃	#18445
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1612

848 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)(54,56,58,62)(55,60,61,67)(57,63,64,68)(59,66,65,69)(70,71)(72,75)(73,77)(74,76)(78,79)(80,84)(81,83)(82,85);(56,62)(60,67)(63,68)(66,69)(71,76)(73,83)(75,84)(79,82);(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)(54,55,58,61)(56,60,62,67)(57,65,64,59)(63,69,68,66)(70,72,74,80)(71,75,76,84)(73,82,83,79)(77,85,81,78);(55,59)(57,64)(60,66)(61,65)(63,68)(67,69)(70,71,74,76)(72,78,80,85)(73,81,83,77)(75,82,84,79)

Polycyclic presentation:

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

Conjugacy classes & character table

635 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 144834);