C₉×#1365 #11766086

order	1152
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#1365

Subgroups

	subgroup	G/H
center	C₂²×C₉	#102
commutator	C₂×C₈	C₂×C₄×C₉
Frattini, cosocle	C₃×C₄×C₈	C₂²×C₃
socle, prime radical	C₂²×C₃	C₃×#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×C₈×C₉ ◁ G

129 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 129 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 1152

A minimal faithful permutation representation — 2 generators on 33 points:

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

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, 97);