C₃×#13661 #10632268

order	768
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13661

Subgroups

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

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

normal subgroups only

This group has 334 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)(10,14)(11,23)(12,26)(13,18)(15,19)(16,20)(17,24)(21,33)(22,29)(25,39)(27,35)(28,34)(30,38)(31,36)(32,37);(9,14)(10,19)(11,20)(12,24)(15,16)(17,33)(18,29)(21,38)(22,36)(23,27)(25,37)(26,28)(30,39)(31,32)(34,35);(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, 6649);