#27390108

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1868 ⋊ C₂
Sylow	C₅₃ ⋊ #21484

Subgroups

	subgroup	G/H
center	C₄	#120×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1868	C₂
socle	C₂×C₅₃	C₄×#120
prime radical	#20284109	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1868	D₅₃
hypocenter	C₅₃	#21484
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1868

1306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

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

Polycyclic presentation:

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

Conjugacy classes & character table

890 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 175584);