#27383963

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1830 ⋊ C₂
Sylow	C₅₃ ⋊ #20713

Subgroups

	subgroup	G/H
center	C₂	#381×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1830	C₂
socle	C₂×C₅₃	C₂×#381
prime radical	C₂²×#11719949	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1830	D₅₃
hypocenter	C₅₃	#20713
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1830

1244 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)(54,58,60,59)(55,63,64,65)(56,68,69,74)(57,71,72,73)(61,76,66,79)(62,77,70,78)(67,81,82,83)(75,84,80,85);(54,57)(55,62,64,70)(56,67)(58,71)(59,73)(60,72)(61,75,66,80)(63,77,65,78)(68,81)(69,82)(74,83)(76,84,79,85);(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)(55,61)(57,67)(58,68)(59,74)(60,69)(62,75)(63,76)(64,66)(65,79)(70,80)(71,81)(72,82)(73,83)(77,84)(78,85);(54,55,59,65,60,64,58,63)(56,66,68,79,69,61,74,76)(57,70,73,77,72,62,71,78)(67,75,81,84,82,80,83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

812 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 169439);