C₉×#2030 #11766751

order	1152
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#2030

Subgroups

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₂⁴	C₂³×C₉
Frattini, prime radical, cosocle	C₂⁴×C₃	C₂³×C₃
socle	C₂²×C₃	C₃×#120
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂⁴×C₉ ◁ G

435 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 126 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)(10,11,13,15)(12,16,17,14)(18,19)(20,23)(21,24)(22,25)(26,31)(27,30)(28,32)(29,33);(10,12,13,17)(11,14,15,16)(18,20)(19,23)(21,26)(22,27)(24,31)(25,30)(28,33)(29,32);(18,21,22,28)(19,24,25,32)(20,26,27,33)(23,29,30,31)

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, 762);