#27486557

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1861 ⋊ C₂
Sylow	C₅₃ ⋊ #29385

Subgroups

	subgroup	G/H
center	C₄	#16811188
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1861	C₂
socle	C₂×C₅₃	#2598
prime radical	#20275781	C₂
hypercenter	#1861	D₅₃
hypocenter	C₅₃	#29385
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1861

662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 272033);