C₃×#10043 #10628650

order	768
nilpotency class	3
isoclinism	isoclinic class of #10038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10043

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂²×C₃×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 476 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) = 192

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

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

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3031);