C₉×#1646 #11766367

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#1646

Subgroups

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₄²	C₂³×C₉
Frattini, cosocle	C₃×C₄²	C₂³×C₃
socle	C₂²×C₃	C₃×#120
prime radical	C₃×#555	C₂×C₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ G

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

normal subgroups only

This group has 318 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) = 33
transitive degree	μ_t(G) = 576

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)(18,19,23,27)(20,28,31,32)(21,26,22,25)(24,30,33,29);(10,11,14,16)(12,15,13,17)(18,20,23,31)(19,24,27,33)(21,30,22,29)(25,32,26,28);(10,12,14,13)(11,15,16,17)(19,25)(21,22)(24,32)(26,27)(28,33)(29,30)

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, 378);