#27416689

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3020 ⋊ C₂
Sylow	C₅₃ ⋊ #23999

Subgroups

	subgroup	G/H
center	C₂	#20278031
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3020	C₂
socle	C₂×C₅₃	#2404
hypercenter	#3020	D₅₃
hypocenter	C₅₃	#23999
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3020

1146 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)(54,57,59,70)(55,61,63,77)(56,66,67,83)(58,73,74,69)(60,75,65,81)(62,79,71,76)(64,80,78,82)(68,85,72,84);(54,56)(55,60)(57,66)(58,72)(59,67)(61,75)(62,78)(63,65)(64,71)(68,74)(69,85)(70,83)(73,84)(76,80)(77,81)(79,82);(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)(54,55)(56,65,67,60)(57,61)(58,62)(59,63)(64,68,78,72)(66,81,83,75)(69,76)(70,77)(71,74)(73,79)(80,85,82,84);(54,55)(56,64)(57,61)(58,71)(59,63)(60,68)(62,74)(65,72)(66,82)(67,78)(69,79)(70,77)(73,76)(75,84)(80,83)(81,85)

Polycyclic presentation:

2^8,53|b^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,g^b=gh,i^b=i^52,e^c=eh,f^c=fh,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, 202165);