#27498575

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2053 ⋊ C₂
Sylow	C₅₃ ⋊ #30439

Subgroups

	subgroup	G/H
center	C₂	#20280922
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2053	C₂
socle	C₂×C₅₃	#2668
prime radical	#20284070	C₂
hypercenter	#2053	D₅₃
hypocenter	C₅₃	#30439
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2053

882 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)(56,68)(58,59)(61,76)(63,64)(66,77)(67,82)(69,70)(71,80)(72,74)(75,85)(78,81)(83,84);(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)(56,67)(58,72)(59,74)(60,73)(61,75)(63,71)(64,80)(65,79)(66,81)(68,82)(69,83)(70,84)(76,85)(77,78);(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)(54,56,60,70)(55,61,65,77)(57,67,73,84)(58,68,59,69)(62,78,79,75)(63,76,64,66)(71,81,80,85)(72,82,74,83);(54,55)(56,66)(57,71)(58,64)(59,63)(60,65)(61,69)(62,72)(67,78)(68,77)(70,76)(73,80)(74,79)(75,84)(81,83)(82,85)

Polycyclic presentation:

2^8,53|b^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,i^b=i^52,f^c=fh,i^c=i^52,e^d=eh,f^d=fh,i^d=i^52

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 284051);