C₃×#13458 #10632065

order	768
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#13458

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₂³×C₃	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

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

normal subgroups only

This group has 324 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)(8,9,11,15)(10,20)(12,26,25,36)(13,19,14,18)(16,32)(17,28,31,29)(21,35)(22,30)(23,37)(24,38)(27,33)(34,39);(4,5)(6,7)(9,15)(10,21,24,23)(16,33,30,34)(17,31)(18,19)(20,35,38,37)(22,39,32,27)(26,36);(5,7)(8,10,13,23,11,24,14,21)(9,16,18,33,15,30,19,34)(12,27,28,32,25,39,29,22)(17,35,36,38,31,37,26,20)

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, 6446);