#27421495

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3027 ⋊ C₂
Sylow	C₅₃ ⋊ #24410

Subgroups

	subgroup	G/H
center	C₂	#20278085
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3027	C₂
socle	C₂×C₅₃	#2410
hypercenter	#3027	D₅₃
hypocenter	C₅₃	#24410
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3027

1322 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:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,57)(55,61)(56,66)(58,71)(59,70)(60,75)(62,69)(63,77)(64,78)(65,79)(67,81)(68,80)(72,84)(73,82)(74,83)(76,85);(54,55)(56,65)(57,61)(58,73)(59,62)(60,72)(63,74)(64,68)(66,79)(67,76)(69,70)(71,82)(75,84)(77,83)(78,80)(81,85);(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)(54,56,59,67)(55,60,62,64)(57,66,70,81)(58,72,74,68)(61,75,69,78)(63,65,73,76)(71,84,83,80)(77,79,82,85);(54,55)(56,64,67,60)(57,69,70,61)(58,63)(59,62)(65,72,76,68)(66,75)(71,82,83,77)(73,74)(78,81)(79,84)(80,85)

Polycyclic presentation:

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

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, 206971);