#27393297

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1911 ⋊ C₂
Sylow	C₅₃ ⋊ #21784

Subgroups

	subgroup	G/H
center	C₂	#20277289
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1911	C₂
socle	C₂×C₅₃	#2308
prime radical	#20288507	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1911	D₅₃
hypocenter	C₅₃	#21784
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1911

1278 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)(54,56)(55,60)(57,65)(58,66)(59,67)(61,74)(62,64)(63,75)(68,85)(69,82)(70,83)(71,73)(72,84)(76,77)(78,79)(80,81);(54,57,59,70)(55,61,62,77)(56,65,67,83)(58,72,73,69)(60,74,64,76)(63,80,78,68)(66,84,71,82)(75,81,79,85);(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)(54,56)(55,60)(57,65)(58,71)(59,67)(61,76)(62,64)(63,79)(66,73)(68,85)(69,84)(70,83)(72,82)(74,77)(75,78)(80,81);(54,55)(56,64,67,60)(57,68,70,80)(58,63)(59,62)(61,69,77,72)(65,81)(66,79,71,75)(73,78)(74,82)(76,84)(83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 178773);