C₃×#13333 #10631940

order	768
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13333

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₂×C₃×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

546 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 192

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

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6321);