C₉×#1663 #11766384

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#1663

Subgroups

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₄²	C₂³×C₉
Frattini, cosocle	C₃×C₄²	C₂³×C₃
socle, prime radical	C₂²×C₃	C₃×#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ G

258 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 90 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)(10,11,16,20)(12,21,23,24)(13,22,17,25)(14,19,15,18)(26,27,31,29)(28,34,37,41)(30,38,40,33)(32,39,35,36);(10,12)(11,17)(13,20)(14,25)(15,22)(16,23)(18,24)(19,21)(26,28,31,37)(27,32,29,35)(30,39,40,36)(33,41,38,34);(10,13,16,17)(11,12,20,23)(14,24,15,21)(18,22,19,25)(26,29,31,27)(28,35,37,32)(30,33,40,38)(34,39,41,36)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 395);