C₉×#1609 #11766330

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

Aliases

direct product

C₉×#1609

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	C₃×#121	C₂²×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

324 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 33
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 33 points:

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

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, 341);