C₉×#1340 #11766061

order	1152
nilpotency class	4
isoclinism	isoclinic class of #1339
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#1340

Subgroups

	subgroup	G/H
center	C₂×C₉	#327
commutator	C₂×C₈	C₂×C₄×C₉
Frattini, cosocle	C₂×C₃×M₁₆	C₂²×C₃
socle	C₂×C₃	C₃×#327
prime radical	C₂³×C₃	C₃×D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₂×C₉×M₁₆ ◁ G

156 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 2 generators on 41 points:

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

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, 72);