#27364769

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1647 ⋊ C₂
Sylow	C₅₃ ⋊ #18942

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1647	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#1647	D₅₃
hypocenter	C₅₃	#18942
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1647

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150245);