C₉×#2098 #11766819

order	1152
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#2098

Subgroups

	subgroup	G/H
center	C₄×C₉	#127
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#399
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂²×C₄×C₉ ◁ G

315 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 41
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5,6,7,8,9)(10,11,14,19,15,20,16,30)(12,21,23,33,25,37,24,17)(13,26,28,36,29,18,31,35)(22,38,27,41,39,34,40,32);(10,12,16,23,15,25,14,24)(11,17,30,21,20,33,19,37)(13,27,31,39,29,40,28,22)(18,34,36,32,26,38,35,41);(10,13)(11,18)(12,22)(14,28)(15,29)(16,31)(17,32)(19,35)(20,26)(21,38)(23,27)(24,40)(25,39)(30,36)(33,41)(34,37)

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 830);