#27499375

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3152 ⋊ C₂
Sylow	C₅₃ ⋊ #30497

Subgroups

	subgroup	G/H
center	C₂	#20280928
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3152	C₂
socle	C₂×C₅₃	#2669
prime radical	#20284063	C₂
hypercenter	#3152	D₅₃
hypocenter	C₅₃	#30497
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3152

980 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)(55,64)(56,57)(59,78)(60,73)(61,77)(62,63)(65,83)(66,81)(67,84)(68,85)(69,75)(70,74)(71,76)(72,79)(80,82);(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)(55,63)(56,58)(59,74)(60,75)(61,76)(62,64)(65,72)(66,85)(67,82)(68,81)(69,73)(70,78)(71,77)(79,83)(80,84);(54,56)(55,62)(57,73)(58,74)(59,70)(60,69)(61,71)(63,81)(64,72)(65,79)(66,68)(67,80)(75,77)(76,78)(82,83)(84,85);(54,55,60,66,61,67,59,65)(56,68,70,62,71,79,69,80)(57,72,75,63,76,85,74,82)(58,64,78,83,77,84,73,81)

Polycyclic presentation:

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

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 284851);