C₉×#1675 #11766396

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#1675

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	C₂×C₃×D₄	C₃×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ G

240 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 108 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)(10,11,13,16)(12,14,17,15)(18,19,22,25)(20,26,29,33)(21,30,27,24)(23,31,32,28);(18,20,27,31,22,29,21,28)(19,23,30,33,25,32,24,26);(11,14)(12,17)(15,16)

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, 407);