#27369366

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1678 ⋊ C₂
Sylow	C₅₃ ⋊ #19396

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1678	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#127×D₅₃	C₂²
hypercenter	#1678	D₅₃
hypocenter	C₅₃	#19396
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1678

686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 77 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)(63,68)(67,75)(69,76)(74,77);(62,64,66,71)(63,67,68,75)(65,73,72,70)(69,77,76,74);(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)(54,55,56,58,57,59,60,61)(62,63)(64,67)(65,69)(66,68)(70,74)(71,75)(72,76)(73,77);(55,58)(56,60)(59,61)(64,70)(65,72)(67,74)(69,76)(71,73)(75,77)

Polycyclic presentation:

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

Conjugacy classes & character table

803 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 154842);