C₃×#13665 #10632272

order	768
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#13665

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₃
prime radical	C₂³×C₃	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂²×C₃×C₁₆ ◁ G

284 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 71
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 71 points:

(1,2,3)(4,5)(6,7)(8,9,14,19)(10,20,24,51)(11,25,30,49)(12,31,34,65)(13,35,33,18)(15,37,40,21)(16,41,44,28)(17,45,47,27)(22,53,54,70)(23,55,36,50)(26,46,60,71)(29,63,62,57)(32,42,66,69)(38,64,67,59)(39,52,48,58)(43,61,56,68);(8,10,14,24)(9,15,19,40)(11,26,30,60)(12,32,34,66)(13,36,33,23)(16,42,44,69)(17,46,47,71)(18,48,35,39)(20,49,51,25)(21,41,37,28)(22,45,54,27)(29,64,62,59)(31,38,65,67)(43,70,56,53)(50,57,55,63)(52,61,58,68);(5,7)

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6653);