C₉×#2194 #11766915

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₉×#2194

Subgroups

	subgroup	G/H
center	C₂×C₉	#447
commutator	C₂×C₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle	C₂×C₃	C₃×#447
prime radical	C₃×#576	C₂×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#131 ◁ G

324 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 96 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)(10,11,16,22)(12,23,26,39)(13,27,31,38)(14,21,15,20)(17,33,36,24)(18,32,37,25)(19,29,34,30)(28,40,41,35);(10,12)(11,17)(13,28)(14,25)(15,32)(16,26)(18,34)(19,37)(20,35)(21,40)(22,36)(23,27)(24,30)(29,33)(31,41)(38,39);(10,12)(11,18)(13,24)(14,32)(15,25)(16,26)(17,34)(19,36)(20,39)(21,23)(22,37)(27,40)(28,30)(29,41)(31,33)(35,38)

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, 926);