C₃×#13413 #10632020

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

Aliases

direct product

C₃×#13413

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₂×C₃×#493	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

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

normal subgroups only

This group has 540 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) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3)(4,5)(6,7)(8,9)(10,17)(11,12)(13,16)(14,18)(15,21)(19,20);(9,12)(10,18)(11,21)(14,20)(15,16)(17,22)(19,23);(5,7)(8,10,11,19,13,20,21,17)(9,14,15,23,16,18,12,22)

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, 6401);