#27359508

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2817 ⋊ C₂
Sylow	C₅₃ ⋊ #18457

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2817	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288944	C₂
hypercenter	#2817	D₅₃
hypocenter	C₅₃	#18457
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2817

954 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)(55,61)(59,66)(60,67)(65,68)(70,74,76,72)(71,79,80,82)(73,81,75,83)(77,78,85,84);(54,57)(55,60)(56,62)(58,64)(59,65)(61,67)(63,69)(66,68)(70,73,76,75)(71,78,80,84)(72,81,74,83)(77,82,85,79);(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)(54,56)(55,59)(57,62)(58,63)(60,65)(61,66)(64,69)(67,68)(70,72,76,74)(71,77,80,85)(73,81,75,83)(78,82,84,79);(54,55)(56,59)(57,60)(58,61)(62,68)(63,66)(64,67)(65,69)(70,71,76,80)(72,79,74,82)(73,78,75,84)(77,83,85,81)

Polycyclic presentation:

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

Conjugacy classes & character table

710 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 144984);