#72224451

order	5760
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72224450
commutator, cosocle	#13088280	C₂²
Frattini	C₄	C₂×S₆
Fitting, hypercenter	C₈	S₆
hypocenter	SL(2,𝔽₉)	D₄
LCS	G ▷ #13088280 ▷ SL(2,𝔽₉)
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

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

normal subgroups only

This group has 388 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) = 80
transitive degree	μ_t(G) = 80

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

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	ΓL(2,𝔽₉)
maximal normal subgroup of	ΓL(2,𝔽₉)

GAP

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

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