C₉×#1346 #11766067

order	1152
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#1346

Subgroups

	subgroup	G/H
center	C₂²×C₉	#99
commutator	C₂²×C₄	C₂×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄⋊C₄	C₂²×C₃
socle	C₂²×C₃	C₃×#99
prime radical	C₂³×C₃	C₃×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂³×C₉ ◁ C₂×C₉×C₄⋊C₄ ◁ G

186 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 60 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)(12,14)(13,16)(15,17)(18,19,21,25,22,24,29,32)(20,26,27,33,28,23,30,31);(11,14)(12,17)(15,16)(18,20,22,28)(19,23,24,26)(21,27,29,30)(25,33,32,31)

Conjugacy classes & character table

207 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 78);