C₉×#1684 #11766405

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#1684

Subgroups

	subgroup	G/H
center	C₂²×C₉	#121
commutator	C₄²	C₂³×C₉
Frattini, cosocle	C₃×C₄²	C₂³×C₃
socle, prime radical	C₂²×C₃	C₃×#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ G

216 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 1152

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

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

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, 416);