#27360273

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1635 ⋊ C₂
Sylow	C₅₃ ⋊ #18538

Subgroups

	subgroup	G/H
center	C₂²	#120×D₅₃
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1635	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288327	C₂
hypercenter	#1635	D₅₃
hypocenter	C₅₃	#18538
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1635

922 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)(55,61,63,62)(56,65,66,67)(60,64,68,69)(70,73,76,72)(71,79,81,84)(74,83,75,82)(77,78,85,80);(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)(71,78)(74,75)(77,84)(79,85)(80,81)(82,83);(54,56)(55,60)(57,65)(58,67)(59,66)(61,64)(62,69)(63,68)(70,72,76,73)(71,77,81,85)(74,83,75,82)(78,84,80,79);(54,55,59,63)(56,64,66,69)(57,62,58,61)(60,67,68,65)(70,71,76,81)(72,79,73,84)(74,80,75,78)(77,83,85,82)

Polycyclic presentation:

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

Conjugacy classes & character table

716 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 145749);