C₃×#10083 #10628690

order	768
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10083

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂²×C₃×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

526 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 39
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3)(8,9,13,17)(10,18,20,23)(11,16,12,15)(14,21,22,19)(24,25,28,31)(26,32,34,39)(27,35,37,30)(29,36,38,33);(4,5)(6,7)(8,10)(9,14)(11,21)(12,19)(13,20)(15,23)(16,18)(17,22)(24,26,28,34)(25,29,31,38)(27,36,37,33)(30,39,35,32);(5,7)(8,10)(9,14)(11,19)(12,21)(13,20)(15,18)(16,23)(17,22)(27,37)(30,35)(32,39)(33,36)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3071);