C₃×#13415 #10632022

order	768
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#13415

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#459
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

436 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 114 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)(8,9,15,21)(10,13,24,31)(11,22,28,38)(12,20,14,18)(16,19,29,23)(17,33,35,27)(25,30,34,39)(26,36,32,37);(8,10,15,24)(9,16,21,29)(11,25,28,34)(12,19,14,23)(13,18,31,20)(17,26,35,32)(22,33,38,27)(30,37,39,36);(5,7)(8,11,12,26,15,28,14,32)(9,17,18,34,21,35,20,25)(10,22,23,37,24,38,19,36)(13,30,29,27,31,39,16,33)

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, 6403);