#27421470

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2008 ⋊ C₂
Sylow	C₅₃ ⋊ #24408

Subgroups

	subgroup	G/H
center	C₂	#20278082
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2008	C₂
socle	C₂×C₅₃	#2410
hypercenter	#2008	D₅₃
hypocenter	C₅₃	#24408
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2008

1364 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

638 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 206946);