C₉×#2222 #11766943

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₉×#2222

Subgroups

	subgroup	G/H
center	C₂×C₉	#466
commutator	C₂×C₈	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₈	C₂³×C₃
socle	C₂×C₃	C₃×#466
prime radical	C₃×#41	C₃×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#131 ◁ G

210 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 73
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 73 points:

(1,2,3,4,5,6,7,8,9)(10,11,17,23)(12,24,30,52)(13,31,34,48)(14,35,37,46)(15,22,16,21)(18,39,43,27)(19,38,45,33)(20,36,47,25)(26,55,58,72)(28,51,29,50)(32,63,64,44)(40,67,68,73)(41,59,42,53)(49,60,71,57)(54,62,65,70)(56,61,66,69);(10,12,25,53,16,29,37,39,17,30,36,59,15,28,14,27)(11,18,35,51,22,42,47,52,23,43,46,50,21,41,20,24)(13,26,54,73,33,57,64,66,34,58,65,67,38,60,32,56)(19,40,62,55,31,61,63,71,45,68,70,72,48,69,44,49);(10,13)(11,19)(12,26)(14,32)(15,38)(16,33)(17,34)(18,40)(20,44)(21,48)(22,31)(23,45)(24,49)(25,54)(27,56)(28,60)(29,57)(30,58)(35,62)(36,65)(37,64)(39,66)(41,69)(42,61)(43,68)(46,70)(47,63)(50,72)(51,55)(52,71)(53,73)(59,67)

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, 954);