#27276008

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1546 ⋊ C₂
Sylow	C₅₃ ⋊ #8421

Subgroups

	subgroup	G/H
center	C₂²	#16810478
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1546	C₂
socle	C₂²×C₅₃	#379
prime radical	#20288498	C₂
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1546	D₅₃
hypocenter	C₅₃	#8421
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1546

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(62,65,68,63)(64,73,76,69)(66,71,67,70)(72,74,77,75);(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)(55,59)(56,57)(60,61)(62,64)(63,69)(65,73)(66,75)(67,74)(68,76)(70,77)(71,72);(54,55,57,59,58,61,56,60)(62,63,66,71,68,65,67,70)(64,72,75,73,76,77,74,69)

Polycyclic presentation:

2^8,53|a^2=f,c^2=g,d^2=h,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fgh,i^b=i^52,e^c=eg,f^c=fg,i^c=i^52

Conjugacy classes & character table

875 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 61484);