#27415509

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3585 ⋊ C₂
Sylow	C₅₃ ⋊ #23883

Subgroups

	subgroup	G/H
center	C₂	#20278025
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3585	C₂
socle	C₂×C₅₃	#2403
hypercenter	#3585	D₅₃
hypocenter	C₅₃	#23883
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3585

1696 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)(55,63)(58,59)(61,66)(62,78)(64,65)(68,74)(70,79)(71,72)(75,81)(76,80)(77,85)(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)(54,57,60,73)(55,62,65,79)(56,67,69,83)(58,72,59,71)(61,75,76,77)(63,78,64,70)(66,81,80,85)(68,84,74,82);(54,56)(55,61)(57,67)(58,74)(59,68)(60,69)(62,77)(63,66)(64,80)(65,76)(70,81)(71,84)(72,82)(73,83)(75,79)(78,85);(54,55)(56,66)(57,70)(58,63)(59,64)(60,65)(61,68)(62,72)(67,75)(69,80)(71,79)(73,78)(74,76)(77,83)(81,82)(84,85)

Polycyclic presentation:

2^8,53|c^2=h,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,e^b=eh,e^c=eh,f^c=fh,i^c=i^52,e^d=eh,f^d=fh

Conjugacy classes & character table

1106 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 200985);