#27416709

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2009 ⋊ C₂
Sylow	C₅₃ ⋊ #24001

Subgroups

	subgroup	G/H
center	C₂	#20278029
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#2009	C₂
socle	C₂×C₅₃	#2404
prime radical	#20284077	C₂
hypercenter	#2009	D₅₃
hypocenter	C₅₃	#24001
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2009

1122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 4 generators on 85 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

638 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 202185);