#27360467

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1559 ⋊ C₂
Sylow	C₅₃ ⋊ #18557

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133187
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1559	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂×#11719945	D₄
hypercenter	#1559	D₅₃
hypocenter	C₅₃	#18557
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1559

900 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)(55,60)(59,68)(61,65)(62,69)(70,73,76,72)(71,79,81,84)(74,83,75,82)(77,78,85,80);(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)(71,78)(74,75)(77,84)(79,85)(80,81)(82,83);(54,56,58,64)(55,59,60,68)(57,66,67,63)(61,69,65,62)(70,72,76,73)(71,77,81,85)(74,83,75,82)(78,84,80,79);(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)(54,55,58,60)(56,62,64,69)(57,65,67,61)(59,66,68,63)(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=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,i^a=i^52,e^b=eg,f^b=fh,e^c=eh,f^c=fh,i^c=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, 145943);