#27374985

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3364 ⋊ C₂
Sylow	C₅₃ ⋊ #19858

Subgroups

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3364	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#16810818	C₂²
hypercenter	#3364	D₅₃
hypocenter	C₅₃	#19858
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3364

580 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)(54,56,59,58)(55,60,62,66)(57,63,65,67)(61,68,69,64)(70,73,75,81)(71,77,78,85)(72,79,74,83)(76,84,82,80);(54,57)(55,61)(56,63)(58,67)(59,65)(60,68)(62,69)(64,66)(70,72,75,74)(71,76,78,82)(73,79,81,83)(77,84,85,80);(54,56,59,58)(55,60,62,66)(57,63,65,67)(61,68,69,64)(70,71,75,78)(72,76,74,82)(73,80,81,84)(77,83,85,79);(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)(57,64)(58,66)(59,62)(61,63)(65,68)(67,69)(71,76)(72,74)(77,84)(78,82)(79,83)(80,85)

Polycyclic presentation:

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

Conjugacy classes & character table

950 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 160461);