D₃×SL(2,𝔽₉) #72228162

order	4320
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

direct product

D₃×SL(2,𝔽₉)

Splittings

derived

C₃×SL(2,𝔽₉) ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×A₆
commutator, hypocenter	C₃×SL(2,𝔽₉)	C₂
Fitting, socle, cosocle	C₂×C₃	C₂×A₆
LCS	G ▷ C₃×SL(2,𝔽₉)
UCS	C₁ ◁ C₂

162 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 10 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) = 83
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 2 generators on 83 points:

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

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

A₅×SL(2,𝔽₉)

GAP

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

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