#72228178

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

Aliases

No aliases for this group.

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

168 conjugacy classes of subgroups; only the 8 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 8 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:

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

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72228181

GAP

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

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