C₉×#1830 #11766551

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#1830

Subgroups

	subgroup	G/H
center	C₂×C₉	#381
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#381
prime radical	C₂³×C₃	C₂²×C₃×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂²×C₄×C₉ ◁ G

354 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 150 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)(10,11,14,19,15,18,28,33)(12,20,22,34,23,16,27,30)(13,24,26,38,25,17,31,32)(21,29,37,40,36,35,39,41);(10,12,15,23)(11,16,18,20)(13,21,25,36)(14,27,28,22)(17,29,24,35)(19,34,33,30)(26,39,31,37)(32,41,38,40);(10,13)(11,17,18,24)(12,21)(14,26)(15,25)(16,29,20,35)(19,32,33,38)(22,37)(23,36)(27,39)(28,31)(30,40,34,41)

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 562);