C₉×#1678 #11766399

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#1678

Subgroups

	subgroup	G/H
center	C₂²×C₉	#121
commutator	C₄²	C₂³×C₉
Frattini, cosocle	C₃×C₄²	C₂³×C₃
socle	C₂²×C₃	C₃×#121
prime radical	C₃×#127	C₂²×C₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ G

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

normal subgroups only

This group has 240 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)(11,14)(12,16)(15,17)(18,19)(20,26)(21,23)(22,30)(24,28)(25,29)(27,33)(31,32);(10,11,12,14,13,15,16,17)(19,23)(24,32)(25,30)(26,33);(18,20,21,27)(19,24,23,32)(22,31,29,28)(25,33,30,26)

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 410);