#27422389

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2050 ⋊ C₂
Sylow	C₅₃ ⋊ #24473

Subgroups

	subgroup	G/H
center	C₂	#20278084
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2050	C₂
socle	C₂×C₅₃	#2410
hypercenter	#2050	D₅₃
hypocenter	C₅₃	#24473
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2050

1314 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)(54,58,61,55)(56,68,70,62)(57,73,76,63)(59,64,60,65)(66,77,79,69)(67,82,84,78)(71,74,81,75)(72,85,83,80);(54,57)(55,63)(56,67)(58,73)(59,74)(60,75)(61,76)(62,78)(64,81)(65,71)(66,80)(68,82)(69,83)(70,84)(72,77)(79,85);(54,56)(55,62)(57,72)(58,68)(59,77)(60,69)(61,70)(63,80)(64,79)(65,66)(67,75)(71,82)(73,85)(74,84)(76,83)(78,81);(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)(54,55,60,64,61,58,59,65)(56,66,69,62,70,79,77,68)(57,71,74,73,76,81,75,63)(67,82,72,85,84,78,83,80)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 207865);