#27299776

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#2140 ⋊ C₂
Sylow	C₅₃ ⋊ #12741

Subgroups

	subgroup	G/H
center	C₂	#20274644
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2140	C₂
socle	C₂×C₅₃	#1786
prime radical	C₂×#16811424	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2140	D₅₃
hypocenter	C₅₃	#12741
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2140

930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(54,56)(55,61)(57,66)(58,68)(59,67)(60,73)(62,76)(63,65)(64,77)(69,81)(70,82)(71,83)(72,80)(74,78)(75,79)(84,85);(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)(55,60)(57,69)(58,70)(61,73)(62,64)(63,74)(65,78)(66,81)(68,82)(71,84)(72,75)(76,77)(79,80)(83,85);(54,55)(56,65,67,61)(57,60)(58,72)(59,63)(62,69)(64,70)(66,78,83,73)(68,77,82,80)(71,74)(75,84)(76,81,79,85)

Polycyclic presentation:

2^8,53|a^2=f,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,g^b=gh,f^c=fh,i^c=i^52

Conjugacy classes & character table

872 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 85252);