#27366008

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1661 ⋊ C₂
Sylow	C₅₃ ⋊ #19065

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1661	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20289111	C₂
hypercenter	#1661	D₅₃
hypocenter	C₅₃	#19065
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1661

920 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

554 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 151484);