#27367381

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1682 ⋊ C₂
Sylow	C₅₃ ⋊ #19214

Subgroups

	subgroup	G/H
center	C₂²	#121×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1682	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#127×D₅₃	C₂²
hypercenter	#1682	D₅₃
hypocenter	C₅₃	#19214
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1682

658 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

800 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 152857);