#72210857

order	2880
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃

non-solvable

Aliases

No aliases for this group.

Splittings

derived

SL(2,𝔽₉) ⋊ C₄

Subgroups

	subgroup	G/H
center, Fitting, hypercenter	C₈	A₆
commutator, hypocenter	SL(2,𝔽₉)	C₄
Frattini, cosocle	C₄	C₂×A₆
socle	C₂	C₄×A₆
prime radical	#13088280	C₂
LCS	G ▷ SL(2,𝔽₉)
UCS	C₁ ◁ C₈

117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	GL(2,𝔽₉), #72224451, #72224454
maximal normal subgroup of	GL(2,𝔽₉), #72224451, #72224454

GAP

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

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