#27357414

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2625 ⋊ C₂
Sylow	C₅₃ ⋊ #18239

Subgroups

	subgroup	G/H
center	C₂²	C₂×#13133185
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2625	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	#20288944	C₂
hypercenter	#2625	D₅₃
hypocenter	C₅₃	#18239
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2625

960 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)(70,74,76,72)(71,79,80,81)(73,82,75,83)(77,84,85,78);(54,57)(55,60)(56,62)(58,64)(59,65)(61,67)(63,69)(66,68)(70,73,76,75)(71,78,80,84)(72,82,74,83)(77,79,85,81);(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)(73,82,75,83)(78,79,84,81);(54,55,58,61)(56,59,63,66)(57,60,64,67)(62,68,69,65)(70,71)(72,81)(73,78)(74,79)(75,84)(76,80)(77,82)(83,85)

Polycyclic presentation:

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

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, 142890);