#27359313

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3227 ⋊ C₂
Sylow	C₅₃ ⋊ #18441

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3227	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20289253	C₂
hypercenter	#3227	D₅₃
hypocenter	C₅₃	#18441
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3227

888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 3392

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

Polycyclic presentation:

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

Conjugacy classes & character table

869 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 144789);