C₉×#2180 #11766901

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#2180

Subgroups

	subgroup	G/H
center	C₂×C₉	#442
commutator	C₈	C₂²×C₄×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle	C₂×C₃	C₃×#442
prime radical	C₃×#135	C₂²×C₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#97 ◁ G

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

normal subgroups only

This group has 213 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)(12,22,24,33)(13,26)(14,18)(15,30)(16,21)(17,23,34,40)(19,27)(20,32)(25,37,31,39)(28,38,41,35)(29,36);(10,12)(11,17)(13,23)(14,28)(15,31)(16,24)(18,33)(19,37)(20,38)(21,34)(22,36)(25,32)(26,39)(27,40)(29,41)(30,35);(10,13,15,14,16,27,32,29)(11,18,20,26,21,36,30,19)(12,23,31,28,24,40,25,41)(17,33,38,39,34,22,35,37)

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, 912);