#27486577

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1881 ⋊ C₂
Sylow	C₅₃ ⋊ #29387

Subgroups

	subgroup	G/H
center	C₄	#16811188
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1881	C₂
socle	C₂×C₅₃	#2598
prime radical	#20275818	C₂
hypercenter	#1881	D₅₃
hypocenter	C₅₃	#29387
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1881

646 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)(55,63,65,64)(57,71)(61,76,77,66)(62,70,79,80)(67,82)(72,74)(75,81,78,85)(83,84);(54,57)(55,62)(56,67)(58,72)(59,74)(60,71)(61,75)(63,80)(64,70)(65,79)(66,81)(68,83)(69,82)(73,84)(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)(54,56,58,68,60,69,59,73)(55,61,64,66,65,77,63,76)(57,67,72,83,71,82,74,84)(62,78,70,85,79,75,80,81);(54,55)(56,66)(57,70)(58,63)(59,64)(60,65)(61,68)(62,72)(67,75)(69,76)(71,80)(73,77)(74,79)(78,82)(81,84)(83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 272053);