C₃×#13664 #10632271

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

Aliases

direct product

C₃×#13664

Subgroups

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

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

normal subgroups only

This group has 324 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) = 39
transitive degree	μ_t(G) = 192

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

(1,2,3)(4,5)(6,7)(8,9,13,18)(10,14,22,29)(11,23,25,39)(12,26,24,17)(15,30,32,19)(16,33,34,20)(21,35,36,28)(27,38,31,37);(9,14)(10,19)(11,20)(12,24)(15,16)(17,35)(18,29)(21,38)(22,30)(23,27)(25,33)(26,28)(31,39)(32,34)(36,37);(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, 6652);