#72224449

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

derived

SL(2,𝔽₉) ⋊ C₂²

Subgroups

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

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

normal subgroups only

This group has 222 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) = 80
transitive degree	μ_t(G) = 80

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

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224451
maximal normal subgroup of	#72224451

GAP

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

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