#27375570

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1625 ⋊ C₂
Sylow	C₅₃ ⋊ #19899

Subgroups

	subgroup	G/H
center	C₂²	#16811255
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1625	C₂
socle	C₂²×C₅₃	D₄²
hypercenter	#1625	D₅₃
hypocenter	C₅₃	#19899
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1625

666 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)(56,64)(57,67)(59,69)(60,66)(62,68)(63,65);(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)(56,63)(59,68)(62,69)(64,65)(71,76)(73,81)(75,82)(77,78)(79,85)(83,84);(54,56)(55,59)(57,64)(58,63)(60,69)(61,68)(62,66)(65,67)(70,72,74,80)(71,75,78,84)(73,79,81,85)(76,82,77,83);(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)(56,62,63,69)(57,66,67,60)(59,64,68,65)(70,71)(72,75)(73,76)(74,78)(77,81)(79,82)(80,84)(83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 161046);