C₉×#2103 #11766824

order	1152
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₉×#2103

Subgroups

	subgroup	G/H
center	C₄×C₉	#124
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#400
prime radical	C₂³×C₃	C₂×C₃×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂²×C₄×C₉ ◁ G

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

normal subgroups only

This group has 222 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) = 41
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5,6,7,8,9)(10,11,14,19,15,20,16,31)(12,21,23,33,25,36,24,17)(13,26,28,40,29,18,30,35)(22,34,37,41,38,39,27,32);(10,12,16,23,15,25,14,24)(11,17,31,21,20,33,19,36)(13,27,30,22,29,37,28,38)(18,34,40,41,26,39,35,32);(10,13,14,28,15,29,16,30)(11,18,19,35,20,26,31,40)(12,22,23,37,25,38,24,27)(17,32,21,34,33,41,36,39)

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, 835);