#27365340

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1661 ⋊ C₂
Sylow	C₅₃ ⋊ #18996

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1661	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288852	C₂
hypercenter	#1661	D₅₃
hypocenter	C₅₃	#18996
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1661

948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 93 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)(62,66,68,67)(63,71,73,72)(64,76,77,80)(65,78,79,81)(69,74,83,86)(70,84,85,87)(75,90,92,91)(82,88,93,89);(54,55)(56,61)(57,59)(58,60)(62,65)(63,70)(64,75)(66,78)(67,81)(68,79)(69,82)(71,84)(72,87)(73,85)(74,88)(76,90)(77,92)(80,91)(83,93)(86,89);(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)(56,60)(58,61)(62,64)(63,69)(65,75)(66,76)(67,80)(68,77)(70,82)(71,74)(72,86)(73,83)(78,90)(79,92)(81,91)(84,88)(85,93)(87,89);(55,58)(56,60)(59,61)(62,63,68,73)(64,74,77,86)(65,70,79,85)(66,72,67,71)(69,80,83,76)(75,89,92,88)(78,87,81,84)(82,90,93,91)

Polycyclic presentation:

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

Conjugacy classes & character table

554 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150816);