C₉×#1617 #11766338

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#1617

Subgroups

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₃×#120
prime radical	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

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

normal subgroups only

This group has 303 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) = 288

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

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

Conjugacy classes & character table

207 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 349);