#27416613

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1912 ⋊ C₂
Sylow	C₅₃ ⋊ #23992

Subgroups

	subgroup	G/H
center	C₂	#20278032
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1912	C₂
socle	C₂×C₅₃	#2404
prime radical	#16811459	C₂²
hypercenter	#1912	D₅₃
hypocenter	C₅₃	#23992
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1912

1128 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)(55,61)(56,65)(58,68)(59,67)(60,73)(62,76)(63,75)(64,78)(66,80)(69,82)(70,83)(71,81)(72,84)(74,79)(77,85);(54,56)(55,60)(57,65)(58,70)(59,66)(61,73)(62,77)(63,74)(64,71)(67,80)(68,83)(69,72)(75,79)(76,85)(78,81)(82,84);(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)(56,64,66,70)(60,72,74,77)(65,78,80,83)(73,84,79,85);(54,55,59,63)(56,60,66,74)(57,61,67,75)(58,69,71,62)(64,77,70,72)(65,79,80,73)(68,82,81,76)(78,84,83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

794 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 202089);