C₃×#13616 #10632223

order	768
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#13616

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×D₁₆
commutator	C₁₆	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₁₆	C₂³×C₃
prime radical	C₂×C₃×#508	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂²×C₃×C₁₆ ◁ G

302 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 384

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

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6604);