C₃×#9983 #10628590

order	768
nilpotency class	3
isoclinism	isoclinic class of #9980
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#9983

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₃×#2815	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

602 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 35
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 2971);