A₅×SL(2,𝔽₉) #72228165

order	43200
isoclinism	isoclinic class of A₅×SL(2,𝔽₉)
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

A₅×SL(2,𝔽₉)

Subgroups

	subgroup	G/H
center, Frattini, Fitting, cosocle, hypercenter	C₂	A₅×A₆
commutator, hypocenter	G	C₁
socle	C₂×A₅	A₆
LCS	G
UCS	C₁ ◁ C₂

359 conjugacy classes of subgroups; only the 6 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 400

A minimal faithful permutation representation — 2 generators on 85 points:

(1,4,12)(2,7,20)(3,10,27)(5,15,36)(6,18,42)(8,22,46)(9,24,44)(11,16,38)(13,31,52)(14,34,61)(17,40,21)(19,32,58)(23,48,72)(25,29,30)(28,47,49)(33,60,37)(35,55,74)(39,65,79)(43,64,66)(51,62,77)(54,73,75)(56,71,57)(59,78,70)(67,69,80)(81,82)(83,84);(1,15,11,44,5,4,35,37)(2,31,19,21,13,7,46,57)(3,53,51,12,14,26,49,36)(6,50,67,20,30,41,66,52)(8,59,71,45,32,23,40,76)(9,29,17,27,33,42,56,61)(10,78,72,77,34,48,70,47)(16,75,24,63,55,39,60,68)(18,54,79,80,25,65,73,64)(22,43,74,28,58,69,38,62)(81,82,83,84,85)

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,4,12)(2,7,20)(3,10,27)(5,15,36)(6,18,42)(8,22,46)(9,24,44)(11,16,38)(13,31,52)(14,34,61)(17,40,21)(19,32,58)(23,48,72)(25,29,30)(28,47,49)(33,60,37)(35,55,74)(39,65,79)(43,64,66)(51,62,77)(54,73,75)(56,71,57)(59,78,70)(67,69,80)(81,82)(83,84), (1,15,11,44,5,4,35,37)(2,31,19,21,13,7,46,57)(3,53,51,12,14,26,49,36)(6,50,67,20,30,41,66,52)(8,59,71,45,32,23,40,76)(9,29,17,27,33,42,56,61)(10,78,72,77,34,48,70,47)(16,75,24,63,55,39,60,68)(18,54,79,80,25,65,73,64)(22,43,74,28,58,69,38,62)(81,82,83,84,85));