#27362087

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1619 ⋊ C₂
Sylow	C₅₃ ⋊ #18704

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1619	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#1619	D₅₃
hypocenter	C₅₃	#18704
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1619

972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 85 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 147563);