#27532674

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2069 ⋊ C₂
Sylow	C₅₃ ⋊ #32870

Subgroups

	subgroup	G/H
center	C₂	#20282809
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2069	C₂
socle	C₂×C₅₃	#2826
prime radical	#20275823	C₂
hypercenter	#2069	D₅₃
hypocenter	C₅₃	#32870
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2069

502 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)(55,62)(56,67)(58,71)(59,72)(60,75)(61,77)(63,79)(64,80)(65,81)(66,82)(68,84)(69,83)(70,73)(74,78)(76,85);(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)(55,61)(56,66)(57,70)(58,73)(60,74)(63,64)(67,83)(68,69)(71,72)(75,81)(77,79)(78,85);(54,56,58,68,59,69,73,66)(55,60,61,65,64,74,63,76)(57,67,70,82,72,83,71,84)(62,78,79,81,80,75,77,85);(54,55,58,63,59,64,73,61)(56,65,68,60,69,76,66,74)(57,62,71,79,72,80,70,77)(67,81,84,75,83,85,82,78)

Polycyclic presentation:

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

Conjugacy classes & character table

626 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 318150);