ΓL(2,𝔽₉) #72224457

order	11520
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

general semilinear

ΓL(2,𝔽₉)

Subgroups

	subgroup	G/H
center, socle	C₂	#72224456
commutator, cosocle	#72210857	C₂²
Frattini, Fitting, hypercenter	C₈	PΓL(2,𝔽₉)
prime radical	#72224451	C₂
hypocenter	SL(2,𝔽₉)	SD₁₆
LCS	G ▷ #72210857 ▷ #13088280 ▷ SL(2,𝔽₉)
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

349 conjugacy classes of subgroups; only the 11 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 11 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) = 80
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 80 points:

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

Conjugacy classes & character table

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