#27360633

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3141 ⋊ C₂
Sylow	C₅₃ ⋊ #18571

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3141	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288742	C₂
hypercenter	#3141	D₅₃
hypocenter	C₅₃	#18571
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3141

1008 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)(56,63)(59,68)(62,69)(64,66)(72,81)(76,85)(80,84)(82,83);(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)(71,77)(73,74)(76,84)(78,79)(80,85)(82,83);(54,56)(55,59)(57,66)(58,63)(60,69)(61,68)(62,65)(64,67)(70,72)(71,76)(73,83)(74,82)(75,81)(77,84)(78,80)(79,85);(54,55,58,61)(56,62,63,69)(57,65,67,60)(59,66,68,64)(70,71)(72,80)(73,78)(74,77)(75,79)(76,82)(81,84)(83,85)

Polycyclic presentation:

2^8,53|a^2=gh,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, 146109);