C₉×#2255 #11766976

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#2255

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂×D₈
commutator	C₂×C₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle, prime radical	C₂²×C₃	C₂×C₃×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×C₈×C₉ ◁ G

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

normal subgroups only

This group has 180 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) = 1152

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

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, 987);