C₉×#2234 #11766955

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₉×#2234

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂×D₈
commutator	C₂×C₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×D₈
prime radical	C₃×#466	C₂×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×C₈×C₉ ◁ G

213 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 90 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)(12,15)(13,14)(16,17)(19,23)(20,26)(22,29)(24,32)(25,28)(27,33)(30,31);(11,14)(15,17)(18,19,21,23)(20,24,27,31)(22,25,28,29)(26,30,33,32);(10,12,13,16)(11,15,14,17)(18,20,21,27)(19,24,23,31)(22,30,28,32)(25,33,29,26)

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, 966);