#27415660

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#3485 ⋊ C₂
Sylow	C₅₃ ⋊ #23898

Subgroups

	subgroup	G/H
center	C₂	#20278025
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#3485	C₂
socle	C₂×C₅₃	#2403
hypercenter	#3485	D₅₃
hypocenter	C₅₃	#23898
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3485

1718 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 201136);