#72209591

order	2160
isoclinism	isoclinic class of #72209591
automorphism	Aut(G) = PΓL(2,𝔽₉) (order 1440) · Inn(G) = A₆ · Out(G) = C₂²
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, Fitting, socle, cosocle, hypercenter	C₂×C₃	A₆
commutator, hypocenter	G	C₁
LCS	G
UCS	C₁ ◁ C₂×C₃

51 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

(1,2,3)(4,6,8)(5,12,9,11,7,13)(10,16,15,18,14,17)(19,20,24,21)(22,32,29,33)(23,34,38,35)(25,39,28,40)(26,43,30,44)(27,46,31,47)(36,67,71,68)(37,72,74,73)(41,76,50,77)(42,56,51,59)(45,79,52,80)(48,64,53,61)(49,82,54,81)(55,88,58,89)(57,75,60,69)(62,93,65,94)(63,70,66,95)(78,97,85,98)(83,96,86,92)(84,90,87,91);(1,4,5)(2,6,7)(3,8,9)(10,11,16)(12,17,14)(13,18,15)(20,25,22)(21,28,29)(23,36,37)(26,45,46)(27,48,49)(30,52,47)(31,53,54)(32,55,56)(33,58,59)(34,44,61)(35,43,64)(38,71,74)(39,42,75)(40,51,69)(41,78,72)(50,85,73)(57,90,83)(60,91,86)(63,93,84)(66,94,87)(67,79,97)(68,80,98)(70,88,96)(89,92,95);(1,2,3)(4,10,11)(5,9,7)(6,14,12)(8,15,13)(19,22,23)(20,26,27)(21,30,31)(24,29,38)(25,41,42)(28,50,51)(32,43,57)(33,44,60)(34,62,63)(35,65,66)(36,69,70)(37,59,52)(45,74,56)(48,58,81)(49,83,84)(53,55,82)(54,86,87)(61,92,79)(64,96,80)(67,78,94)(68,85,93)(71,75,95)(72,88,77)(73,89,76);(19,24)(20,21)(22,29)(23,38)(25,28)(26,30)(27,31)(32,33)(34,35)(36,71)(37,74)(39,40)(41,50)(42,51)(43,44)(45,52)(46,47)(48,53)(49,54)(55,58)(56,59)(57,60)(61,64)(62,65)(63,66)(67,68)(69,75)(70,95)(72,73)(76,77)(78,85)(79,80)(81,82)(83,86)(84,87)(88,89)(90,91)(92,96)(93,94)(97,98)

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209688, #72227488, #72227492
maximal quotient of	#72227492, #72227496

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

gap> G := Group((1,2,3)(4,6,8)(5,12,9,11,7,13)(10,16,15,18,14,17)(19,20,24,21)(22,32,29,33)(23,34,38,35)(25,39,28,40)(26,43,30,44)(27,46,31,47)(36,67,71,68)(37,72,74,73)(41,76,50,77)(42,56,51,59)(45,79,52,80)(48,64,53,61)(49,82,54,81)(55,88,58,89)(57,75,60,69)(62,93,65,94)(63,70,66,95)(78,97,85,98)(83,96,86,92)(84,90,87,91), (1,4,5)(2,6,7)(3,8,9)(10,11,16)(12,17,14)(13,18,15)(20,25,22)(21,28,29)(23,36,37)(26,45,46)(27,48,49)(30,52,47)(31,53,54)(32,55,56)(33,58,59)(34,44,61)(35,43,64)(38,71,74)(39,42,75)(40,51,69)(41,78,72)(50,85,73)(57,90,83)(60,91,86)(63,93,84)(66,94,87)(67,79,97)(68,80,98)(70,88,96)(89,92,95), (1,2,3)(4,10,11)(5,9,7)(6,14,12)(8,15,13)(19,22,23)(20,26,27)(21,30,31)(24,29,38)(25,41,42)(28,50,51)(32,43,57)(33,44,60)(34,62,63)(35,65,66)(36,69,70)(37,59,52)(45,74,56)(48,58,81)(49,83,84)(53,55,82)(54,86,87)(61,92,79)(64,96,80)(67,78,94)(68,85,93)(71,75,95)(72,88,77)(73,89,76), (19,24)(20,21)(22,29)(23,38)(25,28)(26,30)(27,31)(32,33)(34,35)(36,71)(37,74)(39,40)(41,50)(42,51)(43,44)(45,52)(46,47)(48,53)(49,54)(55,58)(56,59)(57,60)(61,64)(62,65)(63,66)(67,68)(69,75)(70,95)(72,73)(76,77)(78,85)(79,80)(81,82)(83,86)(84,87)(88,89)(90,91)(92,96)(93,94)(97,98));