#27498556

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3068 ⋊ C₂
Sylow	C₅₃ ⋊ #30438

Subgroups

	subgroup	G/H
center	C₂	#20280918
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3068	C₂
socle	C₂×C₅₃	#2668
hypercenter	#3068	D₅₃
hypocenter	C₅₃	#30438
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3068

874 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)(54,57,59,72)(55,62,64,70)(56,66,69,83)(58,74,73,71)(60,75,63,79)(61,77,78,84)(65,80,76,85)(67,82,68,81);(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)(54,56)(55,61)(57,66)(58,67)(59,69)(60,65)(62,77)(63,76)(64,78)(68,73)(70,84)(71,81)(72,83)(74,82)(75,80)(79,85);(55,60)(57,71)(58,73)(61,76)(62,70)(63,64)(65,78)(66,81)(67,68)(72,74)(80,85)(82,83);(54,55)(56,65)(57,70)(58,60)(59,64)(61,68)(62,72)(63,73)(66,80)(67,78)(69,76)(71,75)(74,79)(77,81)(82,84)(83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

788 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 284032);