#27365473

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1532 ⋊ C₂
Sylow	C₅₃ ⋊ #19011

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133187
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1532	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288854	C₂
hypercenter	#1532	D₅₃
hypocenter	C₅₃	#19011
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1532

886 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

866 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 150949);