#27365776

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3567 ⋊ C₂
Sylow	C₅₃ ⋊ #19040

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3567	C₂
socle	C₂²×C₅₃	C₂×#120
hypercenter	#3567	D₅₃
hypocenter	C₅₃	#19040
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3567

872 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)(55,62)(60,65)(61,68)(67,69)(70,72,74,80)(71,76,77,78)(73,79,81,85)(75,83,82,84);(54,56)(55,61)(57,63)(58,66)(59,64)(60,67)(62,68)(65,69)(73,81)(75,82)(79,85)(83,84);(54,55,59,62)(56,61,64,68)(57,65,58,60)(63,69,66,67)(70,71,74,77)(72,78,80,76)(73,82,81,75)(79,83,85,84);(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)(55,60)(56,63)(57,58)(62,65)(64,66)(67,69)(71,75)(73,81)(76,83)(77,82)(78,84)(79,85)

Polycyclic presentation:

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

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 151252);