C₉×#2221 #11766942

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#2221

Subgroups

	subgroup	G/H
center	C₂×C₉	#466
commutator	C₂×C₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle	C₂×C₃	C₃×#466
prime radical	C₃×#2221	C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#131 ◁ G

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

normal subgroups only

This group has 222 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) = 288

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

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

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, 953);