C₉×#1347 #11766068

order	1152
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#1347

Subgroups

	subgroup	G/H
center	C₂²×C₉	#102
commutator	C₂×C₈	C₂×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₈	C₂²×C₃
socle	C₂²×C₃	C₃×#102
prime radical	C₃×#466	C₂×C₃
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×C₈×C₉ ◁ G

189 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 288

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

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

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, 79);