#27360875

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3165 ⋊ C₂
Sylow	C₅₃ ⋊ #18593

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3165	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288628	C₂
hypercenter	#3165	D₅₃
hypocenter	C₅₃	#18593
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3165

916 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 6784

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

Polycyclic presentation:

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

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 146351);