#27366545

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3167 ⋊ C₂
Sylow	C₅₃ ⋊ #19127

Subgroups

	subgroup	G/H
center	C₂×C₄	#13133188
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3167	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#20284769	C₂
hypercenter	#3167	D₅₃
hypocenter	C₅₃	#19127
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3167

642 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)(54,57,59,58)(55,61,62,67)(56,64,66,65)(60,63,68,69)(70,73,75,74)(71,77,78,82)(72,80,81,83)(76,84,85,79);(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)(70,72)(71,76)(73,80)(74,83)(75,81)(77,84)(78,85)(79,82);(54,56,58,65,59,66,57,64)(55,60,61,63,62,68,67,69);(54,55)(56,63)(57,61)(58,67)(59,62)(60,65)(64,68)(66,69)(70,71)(72,79)(73,77)(74,82)(75,78)(76,80)(81,84)(83,85)

Polycyclic presentation:

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

Conjugacy classes & character table

1202 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 152021);