#27360766

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1626 ⋊ C₂
Sylow	C₅₃ ⋊ #18583

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1626	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20284774	C₂
hypercenter	#1626	D₅₃
hypocenter	C₅₃	#18583
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1626

916 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)(54,57,59,58)(55,61,62,67)(56,64,66,65)(60,63,68,69)(70,73,75,74)(71,76,79,78)(72,81,83,82)(77,84,85,80);(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)(70,72)(71,77)(73,81)(74,82)(75,83)(76,84)(78,80)(79,85);(54,56,59,66)(55,60,62,68)(57,65,58,64)(61,69,67,63)(71,76)(73,74)(77,84)(78,79)(80,85)(81,82);(54,55)(56,63)(57,61)(58,67)(59,62)(60,65)(64,68)(66,69)(70,71)(72,80)(73,76)(74,78)(75,79)(77,81)(82,85)(83,84)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 146242);