C₉×#2195 #11766916

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

Aliases

direct product

C₉×#2195

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×#578	C₂×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#131 ◁ G

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

normal subgroups only

This group has 309 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) = 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)(12,24,28,52)(13,29,32,49)(14,33,37,47)(15,22,16,21)(18,40,43,26)(19,38,45,31)(20,35,48,36)(25,53,57,70)(27,58,39,46)(30,61,64,44)(34,50,65,42)(41,62,69,73)(51,56,72,55)(54,59,67,68)(60,66,71,63);(10,12)(11,18)(13,25)(14,34)(15,27)(16,39)(17,28)(19,41)(20,46)(21,42)(22,50)(23,43)(24,33)(26,36)(29,59)(30,62)(31,55)(32,57)(35,40)(37,65)(38,56)(44,70)(45,69)(47,52)(48,58)(49,68)(51,60)(53,61)(54,63)(64,73)(66,67)(71,72);(10,13,17,32)(11,19,23,45)(12,25,28,57)(14,30,37,64)(15,38,16,31)(18,41,43,69)(20,44,48,61)(21,49,22,29)(24,51,52,72)(26,54,40,67)(27,56,39,55)(33,60,47,71)(34,62,65,73)(35,66,36,63)(42,68,50,59)(46,70,58,53)

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, 927);